v. 因式分解
English | Chinese | Remark & Example |
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faction | n. 分数 | A fraction is a number of the form $\frac c d$, where c and d are integers and d $\not =$ 0. If both c and d are multiplied by the same nonzero integer (or fractored by a common factor), the resulting fraction will be quivalent to $\frac c d$. |
numerator | n. 分子 | |
denominator | n. 分母 | |
rational number | 有理数 | |
irrational number | 无理数 | |
common denominator | 公分母 | To add the two fractions with different denominators, first find a common denominator, which is a common multiple of the two denominators. Then convert both fractions to equivalent fractions with the same denominator. Finally, add the numerators and keep the common denominator. |
invert | n. v. 取倒数 | To divide one fraction by another, first invert the second fraction (that is, find its reciprocal), then multiply the first fraction by the inverted fraction. |
reciprocal | adj. 互惠的，倒数的，相互的 n. 倒数，相互关联的事物 | |
mixed number | 带分数 | It consists of an integer part and a fraction part, where the fraction part has a value between 0 and 1, such as 4$\frac 3 8$. |
fractional expressions | 分数形式的表达式 (?) | Numbers of the form $\frac c d$, where either c or d is not an integer and d $\not =$ 0, are called fractional expressions. |
English | Chinese | Remark & Example |
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exponent | 指数 | Exponents are used to denote the repeated muliplication of a number by itself. eg0: In the expression $3^4$, 3 is called the base, 4 is called the exponent, and we read the expression as “3 to the fourth power.” eg1: For all nonzero numbers a, $a^0 = 1$. The expression $0^0$ is undefined. |
base | 基数 | When negative numbers are raised to powers, the result may be positive or negative. eg: $(-3)^2 = 9, (-3)^5 = -243$ |
squaring | 平方 | When the exponent is 2, we call the process squaring. |
square root | 平方根 | A square root of a nonnegative number n is a nubmer r such that $r^2 = n$. eg0: All positive numbers have two square roots, one positive and one negative. eg1: The only square root of 0 is 0. eg2: The expression consisting of the square root symbol $\sqrt \ $ placed over a nonnegative number denotes the nonnegative square root. $\sqrt{100} = 10$, while $\pm 10$ are square roots of 100. |
cube root | 立方根 | There are some notable differences between odd order roots and even order roots (in real number system): For odd order roots, there is exactly one root for every number n, even when n is negative. For even order roots, there are exactly two roots for every positive number n and no roots for any negative number n. |
fourth root | 四次方根 |
English | Chinese | Remark & Example |
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decimal | 小数 | 7532.418 $\Rightarrow$ [Thousands/Hundreds/Tens/Ones or Units/Tenths/Hundredths/Thousandths] |
terminate | v. 有限小数（有理数） | The decimal that results from the long division will either terminate, as in $\frac 1 4 = 0.25$, or repeat without end, as in $\frac 1 9 = 0.111…$. |
repeat | v. 无限循环小数（有理数） | One way to indicate the repeating part of a decimal that repeats without end is to use a bar over the digits that repeat. eg: $\frac {15}{14} = 1.0\overline{714285}$ |
irrational numbers | 无理数 | fraction with integers in the numerator and denominator $\Longleftrightarrow$ rational number $\Longleftrightarrow$ terminating or repeating decimal |
English | Chinese | Remark & Example |
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real numbers | 实数 | The set of real numbers consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. |
real number line | 实数线 | The set of real numbers can be represented by a number line called the real number line. |
less than or equal to | 小于等于 | |
greater than or equal to | 大于等于 | |
interval | 区间 | |
endpoint | 区间端点 | |
absolute value | 绝对值 | |
triangle inequality | 三角不等式 | $|r + s| \le |r| + |s|$ |
English | Chinese | Remark & Example |
---|---|---|
ratio | 比 | eg0: s/t, “s to t”, s:t eg1: “r to s to t” |
lowest terms | 最简分数/既约分数 | Like fractions, ratios can be reduced to lowest terms. |
proportion | 比例式 | A proportion is an equation relating two ratios. eg: $\frac{9}{12} = \frac{3}{4}$ |
cross multiplication | 交叉相乘 | To solve a problem involving ratios, you can often write a proportion and solve it by cross multiplication. eg: 9*4=12*3 |
English | Chinese | Remark & Example |
---|---|---|
percent | 百分比 | The term percent means per hundred, or hundredths. |
base | （百分比里的）分母 | |
percent change | 百分比变化量（增长率，衰减率） | When a quantity changes from an initial positive amount to another positive amount, you can compute the amount of change as a percent of the initial amount. This is called percent change. percent increase, percent decrease |
The review of algebra begins with algebraic expressions, equations, inequalities, and functions and then progresses to several examples of applying them to solve real-life word problems. The review of algebra ends with coordinate geometry and graphs of functions as other important algebraic tools for solving problems.
代数：代数表达式，等式/方程，不等式，函数，代数应用，坐标几何，函数图像，代数解题工具
English | Chinese | Remark & Example |
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variable | 变量 | A variable is a letter that represents a quantity whose value is unknown. |
algebra expression | 代数表达式 | An algebraic expression has one or more variables and can be written as a single term or as a sum of terms. eg: $w^3z + 5z^2 - z^2 + 6$ has four terms, and $\frac{8}{n + p}$ has one term. |
term | 式子/（代数式的）项 | |
like term | 同类项（次数相同） | Like terms have the same variables, and the corresponding variables have the same exponents. |
constant term | 常数项 | A term that has no variable is called a constant term. |
coefficient | 项的系数 | A number that is multiplied by variables is called the coefficient of a term. eg: Coefficient of 5/y is 5. |
polynomial | 多项式 | A polynomial is the sum of a finite number of terms in which each term is either a constant term or a product of a coefficient and one or more variables with positive integer exponents. eg: The expression $4x^6 + 7x^5 - 3x + 2$ is a polynomial in one variable, x. The polynomial has four terms. |
degree | 度/次数 | The degree of each term is the sum of the exponents of the variables in the term. The degree of a polynomial is the greatest degree of its terms. eg: Polynomials of degrees 2 and 3 are known as quadratic and cubic polynomials, respectively. |
identity | 相等 | A statement of equality between two algebraic expressions that is true for all possible values of the variables involved is called an identity. eg: $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ |
linear equation | 线性方程 | eg: 3z + 5 - t = 20 is a linear equation in two variables. |
quadratic equation | 二次方程 | eg: 20 x^2 = 10 is a quadratic equation in one variables. |
quadratic, cube, biquadratic/quadruplicate/biquadrate | 二次方，三次方，四乘幂/四倍/双二次 |
English | Chinese | Remark & Example |
---|---|---|
base | 基数 | In the algebraic expression $x^a$, where x is raised to the power a, x is called the base and a is called the exponent. |
exponent | 指数 |
English | Chinese | Remark & Example |
---|---|---|
equation | 方程 | An equation is a statement of equality between two mathematical expressions. |
solution | 方程的解 | The values of the variables that make the equation true are called the solutions of the equation. |
solve an equation | 解方程 | To solve an equation means to find the values of the variables that make the equation true, that is, the values that satisfy the equation. Two equations that have the same solutions are called equivalent equations. |
satisfy the equation | 满足方程 | |
equivalent equation | 具有相同解的方程 | The general method for solving an equation is to find successively simpler equivalent equations so that the simplest equivalent equation makes the solutions obvious. eg: $x + 1 = 2$ and $2x + 2 = 4$ are equivalent equations. |
linear equation | 线性方程 | A linear equation is an equation involving one or more variables in which each terms in the equation is either a constant term or a variable multiplied by a coefficient. None of the variables are multiplied together or raised to a power greater than 1. It is possible for a linear equation to have no solutions. Also, it is possible that what looks to be a linear equation could turn out to be an identity when you try to solve it. |
ordered pair | 有序数对 | A solution of linear equation in two variables is an ordered pair of numbers (x, y) that makes the equation true when the values of x and y substituted into the equation. |
system of equations | 方程组 | A set of equations in two or more variables is called a system of equations. There are two basic methods for solving systems of linear equations, by substitution of by elimination. |
simultaneous equations | 联立方程（方程组里的方程互称） | The equations in the system are called simultaneous equations. |
substitution | 带入消元解法 | In the substitution method, one equation is manipulated to express one variable in terms of the other. Then the expression is substituted in the other equation. |
elimination | 加减消元解法 | In the elimination method, the object is to make the coefficients of one variable the same in both equations so that one variable can be eliminated either by adding the equations together or by subtracting one from the other. |
English | Chinese | Remark & Example |
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quadratic equation | 二次方程 | A quadratic equation in the variable x is an equation that can be written in the form $ax^2 + bx + c = 0$, where a, b, c are real numbers and a $\not =$ 0. Quadratic equations have zero, one or two real solutions. |
quadratic formula | 二次方程求根公式 | One way to find solutions of a quadratic equation is to use the quadratic formula: $\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ |
English | Chinese | Remark & Example |
---|---|---|
inequality | 不等式 | |
solve an inequality | 解不等式 | To solve an inequality means to find the set of all values of the variable that make the inequality true. This set of values is also known as the solution set of an inequality. Two inequalities that have the same solution set are called equivalent inequalities. |
solution set | 不等式解集 | |
equivalent inequalities | 等价的不等式 |
English | Chinese | Remark & Example |
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function | 函数 | eg: f(x) is called the value of f at x. |
domain | 自变量的取值范围（定义域） | The domain of a function is the set of all permissible inputs, that is, all permissible values of the variable x. Without an explicit restriction, the domain is assumed to be the set of all values of x for which f(x) is a real number. |
Translating verbal descriptions into algebraic expressions is an essential initial step in solving word problems.
English | Chinese | Remark & Example |
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product | 乘积 | |
interest | 利息 | |
simple interest | 单利 | Simple interest is based only on the initial deposit, which serves as the amount on which interest is computed, called the princial, for the entire time period. eg: If the amount P is invested at a simple annual interest rate of r percent, then the value V of the investment at the end of t years is given by the formula $\displaystyle V = P (1 + \frac {rt}{100})$. |
principal | 本金 | |
compound interest | 复利 | In the case of compound interest, interest is added to the principal at regular time intervals, such as annually, quarterly, and monthly. Each time interest is added to the principal, the interest is said to be compounded. After each compounding, interest is earned on the new principal, which is the sum of the preceding principal and the interest just added. eg0: If the amount P is invested at an annual interest rate of r percent, compounded annually, then the value V of the investment at the end of t years is given by the formula $\displaystyle V = P (1 + \frac{r}{100})^{t}$. eg1: If the amount P is invested at an annual interest rate of r percent, compounded n times per year, then the value V of the investment at the end of t years is given by the formula $\displaystyle V = P (1 + \frac{r}{100})^{nt}$. |
English | Chinese | Remark & Example |
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coordinate | 坐标 | |
rectangular coordinate system / xy-coordinate system / xy-plane | 直角坐标系 | Two real number lines that are perpendicular to each other and that intersect at their respective zero points define a rectangular coordinate system, often called the xy-coordinate system or xy-plane. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. The point where the two axes intersect is called the origin, denoted by O. The two axes divide the plane into four regions called quadrants. |
x-axis y-axis | x轴 y轴 | a and b are symmetric about the x-axis = a is the reflection of b about the x-axis 关于x轴对称 |
origin | 原点O | about the origin 关于原点 |
quadrant | 象限 | |
x-coordinate y-coordinate | 横坐标 纵坐标 | Each point J in the xy-plane can be identified with an ordered pair (x, y) of real numbers and is denoted by J (x, y). The first number in the ordered pair is called the x-coordinate, and the second number is called the y-coordinate. |
graph of an equation | 方程的图形表示 | The graph of an equation in the variables x and y is the set of all points whose ordered pairs (x, y) satisfy the equation. |
line of symmetry | 对称轴 | y = x is a line of symmetry for the graphs of y = 5x + 6 and x = 5y + 6 |
slope | 斜率 | $\displaystyle \frac{y_2 - y _1}{x_2 - x_1}$, this ratio is often called “rise over run”, where rise is the change in y and run is the change in x. |
y-intercept x-intercept | 纵截距 横截距 | Sometimes the terms x-intercept and y-intercept refer to the actual intersection points. |
parallel | 平行 | Two lines are parallel if their slopes are equal. |
perpendicular | 垂直 | Two lines are perpendicular if their slopes are negative reciprocals of each other. |
intersect | 相交 | |
parabola | 抛物线 | The graph of a quadratic equation of the form $y = ax^2 + bx + c$, where a, b and c are constants and $a \not = 0$, is a parabola. |
vertex | 抛物线顶点 | If a is negative, the parabola opens upward and the vertex is its lowest point. |
circle | 圆 | The graph of an equation of the form $(x - a)^2 + (y - b)^2 = r^2$ is a circle with its center at the point (a, b) and with radius r > 0. |
Notice the fine distinction between the graphs of equations and ones of functions. To graph a function in the xy-plane, you represent each input x and its corresponding output f(x) as a point (x, y), where y = f(x). In other words, you use the x-axis for the input and the y-axis for the output.
English | Chinese | Remark & Example |
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piecewise-defined function | 分段函数 | |
dashed curve | 虚线 | |
reflection | 对称 | |
shifted upward / downward / to the left / to the right | 平移 | |
stretched / dilated / shrunk / contracted | 拉伸/扩大/压缩/缩小 | eg: The graph of ch(x) is the graph of h(x) stretched vertically by a factor of c if c > 1. |
The review of geometry begins with lines and angles and progresses to other plane figures, such as polygons, triangles, quadrilaterals, and circles. The review of geometry ends with some basic three-dimensional figures. Coordinate geometry is covered in the Algebra part.
几何：线，角，平面图形（多边形，三角形，四边形，圆），基本三维图形，坐标几何（代数部分已提及）
English | Chinese | Remark & Example |
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line | 直线 | A line is understood to be a straight line that extends in both directions without ending. |
line segment | 线段 | Given any two points on a line, a line segment is the part of the line that contains the two points and all points between them. The two points are called endpoints. |
endpoints | 端点 | |
congruent line segments | 等长线段 | Line segments that have equal lengths are called congruent line segments. |
midpoint | 中点 | The point that divides a line segment into two congruent line segments is called the midpoint of the line segment. |
length | 长度 | Sometimes the notation AB denotes line segment AB, and sometimes it denotes the length of line segment AB. The meaning of the notation can be determined from the context. |
angles | 角 | When two lines intersect at a point, they form four angles. Each angle has a vertex at the point of intersection of the two lines. Somtimes the angle symbol $\angle$ is used instead of the word “angle.” |
vertex | 顶点 | |
opposite angles, vertical angles | 对顶角 | |
congruent angles | 等角 | Opposite angles have equal measure, and angles that have equal measure are called congruent angles. Hence, opposite angles are congruent. |
perpendicular lines | 垂线 | Two lines that intersect to form four congruent angles are called perpendicular lines, for example k $\perp$ m. Each of the four angles has a measure of 90$^{\circ}$. |
right angle | 直角 | An angle with a measure of 90$^{\circ}$ is called a right angle. A common way to indicate that an angle is a right angle is to draw a small square at the vertex of the angle. |
acute angle obtuse angle | 锐角 钝角 | An angle with measure less than 90$^{\circ}$ is called an acute angle, and an angle with measure between 90$^{\circ}$ and 180$^{\circ}$ is called an obtuse angle. |
parallel lines | 平行线 | Two lines in the same plan that do not intersect are called parallel lines, for example k $\parallel$ m. |
English | Chinese | Remark & Example |
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polygon | 多边形 | A polygon is a closed figure formed by three or more line segments all of which are in the same plane. The line segments are called the sides of the polygon. Each side is joined to two other sides at its endpoints, and the endpoints are called vertices (pl. plural of vertex). If a polygon has n sides, it can be divided into n - 2 triangles. Since the sum of measures of the interior angles of a triangle is 180$^{\circ}$, it follows that the sum of the measures of the interior angles of an n-sided polygon is $(n - 2)(180^{\circ})$. |
side | 边 | |
vertices | 顶点 | |
convex | 凸面的 | convex polygon 凸多边形 |
interior angle exterior angle | 内角 外角 | |
diagonal | 对角线 | |
adjacent | 相邻的 | nonadjacent vertices 非相邻顶点 |
triangle | 三角形 | |
quadrilateral | 四边形 | |
pentagon | 五角形 | |
hexagon | 六角形 | |
octagon | 八角形 | |
regular polygon | 正多边形 | eg: In a regular octagon the measure of each angle is $\frac {1080^{\circ}}{8} = 135^{\circ}$. |
perimeter | 周长 | The perimeter of a polygon is the sum of the lengths of its sides. |
area | 面积 | The area of a polygon refers to the area of the region enclosed by the polygon. |
radius diameter | 半径 直径 |
Every triangle has three sides and three interior angles.
The measures of the interior angles add up to $180^{\circ}$.
The length of each side must be less than the sum of the lengths of the other two sides.
English | Chinese | Remark & Example |
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equilateral triangle | 等边三角形 | A triangle with three congruent sides is called an equilateral triangle. |
isosceles triangle | 等腰三角形 | A triangle with at least two congruent sides is called an isosceles triangle. |
right triangle | 直角三角形 | A triangle with an interior right angle is called a right triangle. The side opposite the right angle is called the hypotenuse; the other two sides are called legs. |
hypotenuse legs | （直角三角形的）斜边，弦 （直角三角形的腰 | |
congruent triangles | 全等三角形 | |
similar triangles | 相似三角形 | |
included angle | 夹角 | |
scale factor of similarity | 相似尺度因子，比例因子 |
English | Chinese | Remark & Example |
---|---|---|
rectangle | 矩形 | A quadrilateral with four right angles is called a rectangle. Opposite sides of a rectangle are parallel and congruent, and the two diagonals are also congruent. |
square | 正方形 | A rectangle with four congruent sides is called a square. |
parallelogram | 平行四边形 | A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In a parallelogram, opposite sides are congruent and opposite angles are congruent. |
trapezoid | 梯形 | A quadrilateral in which at least one pair of opposite sides is parallel is called a trapezoid. Two opposite, parallel sides of the trapezoid are called bases of the trapezoid. |
English | Chinese | Remark & Example |
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circle | 圆 | Given a point O in a plane and a positive number r, the set of points in the plane that are a distance of r units from O is called a circle. The point O is called the center of the circle and the distance r is called the radius of the circle. The diameter of the circle is twice the radius. |
center | 圆心 | |
radius | 半径 | |
diameter | 直径 | |
congruent circles | 等圆 | Two circles with equal radius are called congruent circles. |
chord | 弦 | Any line segment joining two points on the circle is called a chord. The terms “radius” and “diameter” can also refer to line segments: A radius is any line segment joining a point on the circle and the center of the circle, and a diameter is a chord that passes through the center of the circle. |
circumference | 圆周 | The distance around a circle is called the circumference of the circle, which is analogous to the perimeter of a polygon. The ratio of the circumference C to the diameter d is the same for all circles and is denoted by the Greek letter $\pi$, that is, $\frac{C}{d} = \pi$, the value of $\pi$ is approximately 3.14 and can also be approximated by the fraction $\frac{22}{7}$. $C = \pi d = 2\pi r$ |
arc | 弧 | Given any two points on a circle, an arc is the part of the circle containing the two points and all the points between them. Two points on a circle are always the endpoints of two arcs. An arc is frequently identified by three points to avoid ambiguity. |
central angle | 圆心角 | A central angle of a circle is an angle with its vertex at the center of the circle. |
measure of an arc | 弧的量度 | The measure of an arc is the measure of its central angle, which is the angle formed by two radius that connect the center of the circle to the two endpoints of the arc. |
length of an arc | 弧的长度 | The ratio of the length of an arc to the circumference is equal to the ratio of the degree measure of the arc to 360$^{\circ}$. |
area | 面积 | The area of a circle with radius r is equal to $\pi r^2$. |
sector | 扇形 | A sector of a circle is a region bounded by an arc of the circle and two radius. |
area of a sector | 扇形面积 | To find the area of a sector, note that the ratio of the area of a sector of a circle to the area of the entire circle is equal to the ratio of the degree measure of its arc to 360$^{\circ}$. |
tangent | 正切，切线 | A tangent to a circle is a line that lies in the same plane as the circle and intersects the circle at exactly one point, called the point of tangency. If aline is tangent to a circle, then a radius drawn to the point of tangency is perpendicular to the tangent line. |
point of tangency | 切点 | |
inscribed circumscribed | 内接的，内切的/雕刻，题写 外接的，外切的/局限，限定 | A polygon is inscribed in a circle if all its vertices lie on the circle, or equivalently, the circle is circumscribed about the polygon. 内接多边形，外接圆 eg0: It is not always the case that if a triangle is inscribed in a circle, the center of the circle is inside the inscribed triangle. It is also possible for the center of the circle to be outside the inscribed triangle, or on one of the sides of inscribed triangle. eg1: If one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle. A polygon is circumscribed about a circle if each side of the polygon is tangent to the circle, or equivalently, the circle is inscribed in the polygon. 外切多边形，内切圆 |
concentric circles | 同心圆 | Two or more circles with the same center are called concentric circles. |
Basic three-dimensional figures include rectangular solids, cubes, cylinders, spheres, pyramids, and cones. In this section, we look at some properties of rectangular solids and right circular cylinders.
English | Chinese | Remark & Example |
---|---|---|
solids, cubes, cylinders, spheres, pyramids, cones, cuboids | 固体，立方体，圆柱体，球体，金字塔/角锥，圆锥，长方体 | |
rectangular solid, rectangular prism | 长方体，矩形棱镜，直角棱镜 | A rectangular solid, or rectangular prism, has 6 rectangular surfaces called faces. Adjacent faces are perpendicular to each other. Each line segment that is the intersection of two faces is called an edge, and each point at which the edges intersect is called a vertex. There are 12 edges and 8 vertices. The dimensions of a rectangular solid are the length $\ell$, the width $w$, and the height $h$. eg: A rectangular solid with six square faces is called a cube, in which case $\ell = w = h$. |
faces | 面 | |
edge | 边 | |
vertex | 顶点 | |
volume | 体积/音量 | The volume V of a rectangular solid is the product of its three dimensions, or $V = \ell w h$. |
surface area | 表面积 | The surface area A of a rectangular solid is the sum of the areas of the six faces, or $A = 2(\ell w + \ell h + wh)$. |
circular cylinder | 圆柱体 | A circular cylinder consists of two bases that are congruent circles lying in parallel planes and a lateral surface made of all line segments that join points on the two circles and that are parallel to the line segment joining the centers of the two circles. The latter line segment is called the axis of the cylinder. |
lateral surface | 侧面 | |
axis | 轴线 | |
right circular cylinder | 正圆柱体 | A right circular cylinder is a circular cylinder whose axis is perpendicular to its bases. The height of a right circular cylinder is the perpendicular distance between the two bases, which is equal to the length of the axis. $V = \pi r^2 h, \ \ A = 2(\pi r^2) + 2\pi r h$ |
height | 高 |
The review of data analysis begins with methods for presenting data, followed by counting methods and probability, and then progresses to distributions of data, random variables, and probability distributions. The review of data analysis ends with examples of data interpretation.
数据分析：数据表示，数数，概率，数据分布，随机变量，概率分布，数据演绎
English | Chinese | Remark & Example |
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quantitative, numerical | 定量的，数字的 | |
categorical, nonnumerical | 分类的，非数字的 | |
distribution of a variable, distribution of data | 变量的分布，数据的分布 | Data are collected from a population by observing one or more variables. The distribution of a variable, or distribution of data, indicates how frequently different categorical or numerical data values are observed in the data. |
frequency, count | 频率（频数） | The frequency, or count, of a particular category or numerical value is the number of times that the category or numerical value appears in the data. |
frequency distribution | 频率（频数）分布 | A frequency distribution is a table or graph that presents the categories or numerical values along with their corresponding frequencies. |
relative frequency | 相对频率（频率） | The relative frequency of a category or a numerical value is the corresponding frequency divided by the total number of data. Relative frequencies may be expressed in terms of percents, fractions, or decimals. |
relative frequency distribution | 相对频率（频率）分布 | A relative frequency distribution is a table or graph that presents the relative frequencies of the categories or numerical values. |
bar graph, bar chart | 条形图 | In a bar graph, each of the data categories or numerical values is represented by a rectangular bar, and the height of each bar is proportional to the corresponding frequency or relative frequency. All of the bars are drawn with the same width, and the bars can be presented either vertically or horizontally. |
segmented bar graph, stacked bar graph | 分段条形图，堆积条形图 | A segmented bar graph, or stacked bar graph, is similar to a regular bar graph except that in a segmented bar graph, each rectangular bar is divided, or segmented, into smaller rectangles that show how the variables is “separated” into other related variables. |
histogram | 直方图，柱状图 | Histograms are graphs of frequency distributions that are similar to bar graphs, but they must have a number line for the horizontal axis, which represents the numerical variable. Also, in a histogram, there are no regular spaces between the bars. Any spaces between bars in a histogram indicate that there are no data in the intervals represented by the spaces. Histograms are useful for identifying the general shape of a distribution of data. |
circle graphs, pie charts | 饼图 | Circle graphs, often called pie charts, are used to represent data that have been separated into a small number of categories. They illustrate how a whole is separated into parts. |
sector | 扇形，扇区 | Each part of a circle graph is called a sector. |
scatterplot | 散点图 | A scatterplot is a type of graph that is useful for showing the relationship between two numerical variables whose values can be observed in a single population of individuals or objects. |
trend | 趋势，走向，倾向 | A scatterplot makes it possible to observe an overall pattern, or trend, in the relationship between the two variables. Also, the strength of the trend as well as striking deviations from the trend are evident. The trend line can be used to make predictions. |
line graph | 线图 | A line graph is another type of graph that is useful for showing the relationship between two numerical variables, especially if one of the variables is time. There is at most one data point for each value on the horizontal axis, similar to a function. The data points are in order from left to right, and consecutive data points are connected by a line segment. |
time series | 时间序列 | When one of the variables is time, it is associated with the horizontal axis, which is labeled with regular time intervals. |
English | Chinese | Remark & Example |
---|---|---|
statistics, statistical measures | 统计量，统计度量 | Data can be described numerically by various statistics, or statistical measures. These statistical measures are often grouped in three categories: measures of central tendency, measures of position, and measures of dispersion. |
central tendency | 集中趋势（中心趋势，指均数、中数、众数） | Measures of central tendency indicate the “center” of the data along the number line and are usually reported as values that represent the data. There are three common measures of central tendency: 1. the arithmetic mean/average/mean; 2. the median; 3. the mode. |
arithmetica mean / average / mean | 算术平均数 | |
median | 中位数 | The median is a measure of central tendency that is fairly unaffected by unusually high or low values relative to the rest of the data. |
mode | 众数 | The mode of a list of numbers is the number that occurs most frequently in the list. |
weighted mean | 加权平均数 | |
weight | 权重 | |
position | 位置，分位 | The three most basic positions, or locations, in a list of numerical data ordered from least to greatest are the beginning, the end, and the middle. It is useful here to label these as L for the least, G for the greatest, and M for the median. Aside from these, the most common measures of position are quartiles and percentiles. As with the mean and the median, the quartiles and percentiles may or may not themselves be values in the data. |
quartiles | 四分位数 | There are three quartile numbers, called the first quartile ($Q_1$) , the second quartile ($Q_2$) , and the third quartile ($Q_3$) , that divide the data into four roughly equal groups. There are various rules to determine the exact values of $Q_1$ and $Q_3$, and the most common rule is that $Q_1$ is the median of the first half of the data in the ordered list and $Q_3$ is the median of the second half of the data in the ordered list. eg: 4 is in the first quartile. The phase “in a quartile” refers to being in one of the four groups determined by $Q_1$, $Q_2$, and $Q_3$. |
percentiles | 百分位数 | There are 99 percentile numbers that divide the data into 100 roughly equal groups ($P_1, P_2, P_3, … , P_{99}$) . Consequently, $Q_1 = P_{25}, M = Q_2 = P_{50}, Q_3 = P_{75}$. |
dispersion | 散布，离差，差量 | Measures of dispersion indicate the degree of spread of the data. Tho most common statistics used as measures of dispersion are the range, the interquartile range, and the standart deviation. |
range | 极差 | The range of the numbers in a group of data is the difference between the greatest number G in the data and the least number L in the data; that is, G - L. |
outlier | 离群值，异常值 | Outliers are unusually small or unusually large in comparison with the rest of the data. |
interquartile | 四分位数，四分点 | |
interquartile range | 四分位数间距，四分位差 | A measure of dispersion that is not usually affected by outliers is the interquartile range. It is defined as the difference between the third quartile and the first quartile; that is, $Q_3 - Q_1$. |
boxplot， box-and-whisker plot | 箱线图，箱形图，盒须图 | $L, Q_1, Q_2, Q_3, G$ can be plotted along a number line to show where the four quartile groups lie. Such plots are called boxplots or box-and-whisker plots, because a box is used to identify each of the two middle quartile groups of data, and “whiskers” extend outward from the boxes to the least and greatest values. |
standard deviation (population standard deviation) | 标准差 | The standard deviation is a measure of spread that depends on each number in the list. Using the mean as the center of the data, the standard deviation takes into account how much each value differs from the eman and then takes a type of average of these differences. |
sample standard deviation | 样本标准差 | The sample standard deviation is qualified with the word “sample” and is computed by dividing the sum of the squared differences by n - 1 instead of n. |
standardization | 标准化 | The process of subtracting the mean from each value and then dividing the result by the standard deviation is called standardization. Standardization is a useful tool because for each data value, it provides a measure of position relative to the rest of the data independently of the variable for which the data was collected and the units of the variable. (?) |
Measures of Central Tendency: P140
Measures of Position: P142
Measures of Dispersion: P143
Example 4.2.9: P146 (Calculating standard deviations)
Fact about standard deviation: In any group of data, most of the data are within 3 standard deviations of the mean.
Thus, when any group of data are standardized, most of the data are transformed to an interval on the number line centered about 0 and extending from -3 to 3. The mean is always transformed to 0.
English | Chinese | Remark & Example |
---|---|---|
set | 集合 | The term set has been used informally in this review to mean a collection of objects that have some property. |
members, elements | 元素 | The objects of a set are called members or elements. |
finite | 有限的（集合） | Some sets are finite, which means that their members can be completely counted. eg: For any finite set S, the number of elements of S is denoted by |S|. |
infinite | 无限的（集合） | Sets that are not finite are called infinite sets, such as the set of all integers. |
empty set | 空集 | A set that has no members is called the empty set and is denoted by the symbol $\varnothing$. |
nonempty | 非空集 | A set with one or more members is called nonempty. |
subset | 子集 | If A and B are sets and all of the members of A are also members of B, then A is a subset of B. |
list | 表 (?) | A list is like a finite set, having members that can all be listed, but with two differences. In a list, the members are ordered-that is, rearranging the members of a list makes it a different list. Thus, the terms “first element,”“second element,” and so on, make sense in a list. Also, elements can be repeated in a list and the repetitions matter. |
intersection | 交集 | The intersection of S and T is the set of all elements that are in both S and T and is denoted by $S \cap T$. |
union | 并集 | The union of S and T is the set of all elements that are in either S or T or both and is denoted by $S \cup T$. |
disjoint / mutually exclusive | 互斥，不相交 | If sets S and T have no elements in common, they are called disjoint or mutually exclusive. |
Venn diagram | 文氏图 | In a Venn diagram, sets are represented by circular regions that overlap if they have elements in common but do not overlap if they are disjoint. |
universal set | 全集 | Sometimes the circular regions are drawn inside a rectangular region, which represents a universal set, of which all other sets involved are subsets. |
inclusion-exclusion principle | 容斥原理 | The number of elements in the union of two sets equals the sum of their individual numbers of elements minus the number of elements in their intersection. | A $\cup$ B | = | A | + | B | - | A $\cap $ B | |
multiplication principle | 乘法原理 | Suppose there are two choices to be made sequentially and that the second choice is independent of the first choice. Suppose also that there are k different possibilities for the first choice and m different possibilities for the second choice. The multiplication principle states that under those conditions, there are km different possibilities for the pair of choices. |
permutation | 排列 | The number of ways to order the n objects is equal to the product $n(n-1)(n-2)…(3)(2)(1)$. Each order is called a permutation, and the product above is called the number of permutations of n objects. |
factorial | 阶乘 | Because products of the form $n(n-1)(n-2)…(3)(2)(1)$ occur frequently when counting objects, a special symbol $n!$, called n-factorial, is used to denote this product. |
permutations of n objects taken k at a time | n中选k的排列数 | $\displaystyle P_n^k =\ _nP_k = \frac{n!}{(n-k)!}$ |
combination | 组合 | selecting without order $(\text{number of ways to select without order}) = \frac{\text{number of ways to select wit order}}{\text{(number of ways to order)}}$ |
combinations of n objects taken k at a time / n choose k | n中选k的组合数 | $\displaystyle C_n^k =\ _nC_k = \binom{n}{k} = \frac{n!}{k!(n-k)!}$ eg0: n choose 0 is 1 eg1: n choose n is 1 eg2: n choose k = n choose (n-k) |
English | Chinese | Remark & Example |
---|---|---|
probability experiment, random experiment | 概率实验，随机实验 | A probability experiment, also called a random experiment, is an experiment for which the result, or outcome, is uncertain. We assume that all of the possible outcomes of an experiment are known before the experiment is performed, but which outcome will actually occur is unknown. |
outcome | 结果，输出，取值 | |
sample space | 样本空间 | The set of all possible outcomes of a random experiment is called the sample space, and any particular set of outcomes is called an event. |
event | 事件 | eg0: The event that both E and F occur, that is, outcomes in the set $E \cap F$. eg1: The event that E or F, or both, occur, that is, outcomes in the set $E \cup F$. eg2: It is common to use the shorter notation “E and F” instead of “both E and F occur” and use “E or F” instead of “E or F or both occur.” |
probability | 概率 | The probability of an event is a number from 0 to 1, inclusive, that indicates the likelihood that the event occurs when the experiment is performed. eg0: If an event E is certain to occur, then P(E) = 1. eg1: If an event E is certain not to occur, then P(E) = 0. eg2: If an event E is possible but not certain to occur, then 0 < P(E) < 1. eg3: The probability that an event E will not occur is equal to 1 - P(E). eg4: If E is an event, then the probability of E is the sum of the probabilities of the outcomes in E. eg5: The sum of the probabilities of all possible outcomes of an experiment is 1. eg6: P(either E or F, or both, occur) = P(E) + P(F) - P(both E and F occur), which is the inclusion-exclusion principle applied to probability. eg7: If E and F are mutually exclusive, then P(both E and F occur) = 0, and therefore, P(either E of F, or both, occur) = P(E) + P(F). eg8: E and F are said to be independent if the occurrence of either event does not affect the occurrence of the other. If two events E and F are independent, then P(both E and F occur) = P(E)P(F). eg9: Note that if P(E) $\not =$ 0 and P(F) $\not =$ 0, then events E and F cannot be both mutually exclusive and independent. For if E and F are independent, then P(both E and F occur) = P(E)P(F) $\not =$ 0, but if E and F are mutually exclusive, then P(both E and F occur) = 0. |
random selection | 随机选择，随机抽样 | The assumption of random selection means that each of the names is equally likely to be selected. |
equally likely | 等可能，等概率 | In general, for a random experiment with a finite number of possible outcomes, if each outcome is equally likely to occur, then the probability that an event E occurs is defined by $\displaystyle P(E) = \frac{\text{the number of outcomes in the event E}}{\text{the number of possible outcomes in the experiment}}$ |
mutually exclusive | 互斥 | Events that cannot occur at the same time are said to be mutually exclusive. |
independent | 独立 | (Notice the difference between “mutually exclusive” and “independent”. “Mutually exclusive” means there is only one experiment with many outcomes. “Independent” means two experiments does not affect each other.) |
In data analysis, variables whose values depend on chance play an important role in linking distributions of data to probability distributions. Such variables are called random variables.
English | Chinese | Remark & Example |
---|---|---|
class | 类 | eg: The measurements were grouped into 50 intervals, or classes, of 10 hours each. |
distribution curve, density curve, frequency curve | 分布曲线，密度曲线，频率曲线 | The distribution can be modeled by a smooth curve that is close to the tops of the bars. Such a model retains the shape of the distribution but is independent of classes. The area under the curve that models the distribution is 1. This model curve is called a distribution curve, but it has other names as well, including density curve and frequency curve. The purpose of the distribution curve is to give a good illustration of a large distribution of numerical data that does not depend on specific classes. |
random variable | 随机变量 | Given a distribution of data, a variable, say X, may be used to represent a randomly chosen value from the distribution. Such a variable X is an example of a random variable, which is a variable whose value is a numerical outcome of a random experiment. The concept of a random variable is more general than representing a randomly chosen value from a distribution of data. A random variable can be any quantity whose value is the result of a random experiment. The possible values of the random variable are the same as the outcomes of the experiment. So any random experiment with numerical outcomes naturally has a random variable associated with it. |
probability distribution | 概率分布 | A fundamental link between data distributions and probability distributions: For a random variable that represents a randomly chosen value from a distribution of data, the probability distribution of the random variable is the same as the relative frequency distribution of the data. |
mean of the random variable X / expected value | 随机变量X的均值/数学期望，预期值 | Another name for the mean of a random variable is expected value. The mean of the random variable X is the sum of the products XP(X) for all values of X, that is, the sum of each value of X multiplied by its corresponding probability P(X). |
discrete random variables | 离散随机变量 | Whose values consist of discrete points on a number line. Fundamental Link: In a histogram representing the probability distribution of a random variable, the area of each bar is proportional to the probability represented by the bar. |
uniform distribution | 均匀分布 | Each of the bars in the histogram of the probability distribution would have the same height. Such a flat histogram indicates a uniform distribution, since the probability is distributed uniformly over all possible outcomes. |
approximately normally distributed | 近似正态分布 | Many natural processes yield data that have a relative frequency distribution shaped somewhat like a bell. Such data are said to be approximately normally distributed and have four properties. Property 1: The mean, median, and mode are all nearly equal. Property 2: The data are grouped fairly symmetrically about the mean. Property 3: About two-thirds of the data are within 1 standard deviation of the mean. Property 4: Almost all of the data within 2 standard deviations of the mean. |
continuous probability distribution | 连续概率分布 | The region below a distribution curve represents a distribution called a continuous probability distribution. There are many different continuous probability distributions, but the most important one is the normal distribution, which has a bell-shaped curve. The area of the region under the curve is 1, and the areas of vertical slices of the region, like the areas of the bars of a histogram, are equal to probabilities of a random variable associated with the distribution. Such a random variable is called a continuous random variable. |
normal distribution | 正态分布 | The properties listed above for the approximately normal distribution of data hold for the normal distribution, except that the mean, median, and mode are exactly the same and the distribution is perfectly symmetric about the mean. A normal distribution, though always shaped like a bell, can be centered around any mean and can be spread out to a greater or lesser degree, depending on the standard deviation. The less the standard deviation, the less spread out the curve is; that is to say, at the mean the curve is higher and as you move away from the mean in either direction it drops down toward the horizontal axis faster. |
continuous random variable | 连续随机变量 | Continuous random variable plays the same role as a random variable that represents a randomly chosen value from a distribution of data. The main difference is that we seldom consider the event in which a continuous random variables is equal to a single values like X=3; rather, we consider events that are described by intervals of values such as 1 < X < 3 and X > 10. |
standard normal distribution | 标准正态分布 | The standard normal distribution is a normal distribution with a mean of 0 and standard deviation equal to 1. To transform a normal distribution with a mean of m and a standard deviation of d to a standard normal distribution, you standardize the values; that is, you subtract m from any observed value of the normal distribution and then divide the result by d. eg0: 1 deviation from the mean: P = 0.683 eg1: 3 deviation from the mean: P = 0.9987 |
算术：整数，分数，小数，实数集，加减乘除，指数和开方，比例和百分数
代数：代数表达式，等式/方程，不等式，函数，代数应用，坐标几何，函数图像，代数解题工具
几何：线，角，平面图形（多边形，三角形，四边形，圆），基本三维图形，坐标几何（代数部分已提及）
数据分析：数据表示，数数，概率，数据分布，随机变量，概率分布，数据演绎
Appendix 0. 多边形
Triangle 三角形
rectangle 矩形
square 正方形
quadrilateral/quadrangle/tetragon 四边形 (later是边，angle是角)
pentagon 五边形
hexagon 六边形
septangle/heptagon/sepilateral 七边形
octagon 八边形
decagon 十边形
hexadecagon 十六边形
Appendix 1. 英语数字前缀
English | Latin | Greek | Example |
---|---|---|---|
one | uni | mono | unique, uniform, unipolar(单极的); monologue(独白), monogamy, monochrome, monopolize(垄断，独占), monotone(单调音，单色调) |
two | du/bi | dis/dy/di | bilingual(双语的), bilateral(双边的), bimonthly; dioxide, disyllable(双音节词), digraph(双字一音；有向图) |
three | tri | tri | triangle, tricycle, trilogy(三部曲) |
four | quadr/quart | tetra | quadrangle(四边形), quadruped(四足动物), quadruple(四倍), quarter(四分之一), quartet(四重奏); tetragon(四边形), tetrahedron(四面体), tetralogy(四部曲) |
five | quint | penta | quintuple(五倍), quintet(五重奏), quincentenary(五百周年纪念的); pentagon(五角大楼), pentathlon(五项全能), pentad(五个一组；五价元素) |
six | sex(t) | hexa | sexangle, sextet(六重奏), sexfoil(六折的); hexagon, hexahedron(六面体), hexapod(六足虫，昆虫) |
seven | sept | hept(a) | septangle, septennial(七年一度的), septavalent(七价的); heptagon, heptahedron, heptachord(七弦琴) |
eight | octo | octo | octopus(章鱼), octapod(章鱼类生物), octuple(八倍), octagon, octet(八人合唱团), octahedron |
nine | nov/non | ennea | nonagon, nonet(九重奏), nonuple(九倍); ennead(九个一组的), enneagon, enneasyllable |
ten | dec | deca/deka | **(deci-, 十分之一) ** decimalism(十进制), decimeter, decigram, decibel(分贝), decimate(取十分之一；大量毁灭，严重破坏); decade(十年), decagon, decaliter(十升), decathlon(十项全能运动) |
half | semi | hemi | semicircle, semicolonial(半殖民地的，colony殖民地), semiconductor, semi-finals; hemisphere, hemicycle, hemiplegia(偏瘫) |
one hundred | cent | hect(o) | (centi-, 百分之一) century, centigrade(摄氏度), centimeter; hectogram(百克), hectoliter(百升), hectometer(百米) |
one thousand | mill | kilo | (milli-, 千分之一) millennium(千年，千周年纪念日), milligram(毫克), millipede(马陆，千足虫), millennial(一千年的，千禧年); kilowatt, kilometer |
ten hundred thousand | mega | mega | megawatt, megaton(百万吨) |
many | multi | poly | multitude(多数，大量), multimedia, multifunctional(多功能的); polycentric(多元的), polygon(多边形), polyglot(通晓多种语言的) |
English | Chinese |
---|---|
single | 1 |
double | 2 |
triple | 3 |
quadruple | 4 |
quintuple | 5 |
sextuple | 6 |
septuple | 7 |
octuple | 8 |
nonuple | 9 |
decuple | 10 |
deci | 1/10 |
centi | 1/100 |
milli (eg: millimeter) | 1/1000 |
Appendix 2. 参考文献
https://zhuanlan.zhihu.com/p/22890294
官方quantitative复习资料传送：http://www.ets.org/s/gre/pdf/gre_math_review.pdf
https://www.ets.org/gre/khan
英语数字前缀：https://wenku.baidu.com/view/9d309a60a98271fe910ef946.html
盒须图：https://en.wikipedia.org/wiki/Box_plot
[TOC]
The review of arithmetic begins with integers, fractions, and decimals and progresses to the set of real numbers. The basic arithmetic operations of addition, subtraction, multiplication, and division are discussed, along with exponents and roots. The review of arithmetic ends with the concepts of ratio and percent.
算术：整数，分数，小数，实数集，加减乘除，指数和开方，比例和百分数
English | Chinese | Remark & Example |
---|---|---|
factor, divisor | n. 因子 v. 因式分解 | When integers are multiplied, each of the multiplied integers is called a factor or divisor of the resulting product. eg: –What are factors of 25? –1,5,25,-1,-5,-25 (notice the negative number) |
multiple divisible | n. 倍数 adj. 可除尽的 | eg0: 60 is a multiple of each of its factors and that 60 is divisible by each of its divisors. eg1: The list of positive multiples of 25 has no end: 25, 50, 75, 100, … eg2: 1 is a factor of every integer, while 0 is a multiple of every integer. 1 is not a multiple of any integer except 1 and -1, while 0 is not a factor of any integer except 0. |
least common multiple | 最小公倍数 | The least common multiple of two nonzero integers c and d is the least positive integer that is a multiple of both c and d. |
greatest common factor, greatest common divisor | 最大公约数 | The greatest comon divisor of two nonzero integers c and d is the greatest positive integer that is a divisor of both c and d. |
quotient | n. 商 | If d is not a divisor of c, the result can be viewed as a fraction or as a decimal, or as a quotient with a remainder. eg: The result of 19 divided by 7 is the quotient 2 with remainder 5, or simply “2 remainder 5”. |
remainder | n. 余数 | eg0: 100 divided by 3 is 33 remainder 1. eg1: -32 divided by 3 is -11 remainder 1, since the greatest multiple of 3 that is less than or equal to -32 is (-11)(3), or -33, which is 1 less than -32. (notice the negative number) eg2: -13 = (-3)(5) + 2 |
even integer | 偶数 | If an integer is divisible by 2, it is called an even integer; otherwise, it is an odd integer. |
odd integer | 奇数 | |
prime number | 质数 | A prime number is an integer than 1 that has only two positive divisors: 1 and itself. eg: The integer 1 is not a prime number, and the integer 2 is the only number that is even. |
prime divisors | 质因数 | Every integer greater than 1 either is a prime number or can be uniquely expressed as a product of factors that are prime numbers, or prime divisors. Such an expression is called a prime factorization. |
prime factorization | 质数分解 | eg: $12 = (2^3)(3)$ |
composite number | 合数 | An integer than 1 that is not a prime number is called a composite number. |
English | Chinese | Remark & Example |
---|---|---|
faction | n. 分数 | A fraction is a number of the form $\frac c d$, where c and d are integers and d $\not =$ 0. If both c and d are multiplied by the same nonzero integer (or fractored by a common factor), the resulting fraction will be quivalent to $\frac c d$. |
numerator | n. 分子 | |
denominator | n. 分母 | |
rational number | 有理数 | |
irrational number | 无理数 | |
common denominator | 公分母 | To add the two fractions with different denominators, first find a common denominator, which is a common multiple of the two denominators. Then convert both fractions to equivalent fractions with the same denominator. Finally, add the numerators and keep the common denominator. |
invert | n. v. 取倒数 | To divide one fraction by another, first invert the second fraction (that is, find its reciprocal), then multiply the first fraction by the inverted fraction. |
reciprocal | adj. 互惠的，倒数的，相互的 n. 倒数，相互关联的事物 | |
mixed number | 带分数 | It consists of an integer part and a fraction part, where the fraction part has a value between 0 and 1, such as 4$\frac 3 8$. |
fractional expressions | 分数形式的表达式 (?) | Numbers of the form $\frac c d$, where either c or d is not an integer and d $\not =$ 0, are called fractional expressions. |
English | Chinese | Remark & Example |
---|---|---|
exponent | 指数 | Exponents are used to denote the repeated muliplication of a number by itself. eg0: In the expression $3^4$, 3 is called the base, 4 is called the exponent, and we read the expression as “3 to the fourth power.” eg1: For all nonzero numbers a, $a^0 = 1$. The expression $0^0$ is undefined. |
base | 基数 | When negative numbers are raised to powers, the result may be positive or negative. eg: $(-3)^2 = 9, (-3)^5 = -243$ |
squaring | 平方 | When the exponent is 2, we call the process squaring. |
square root | 平方根 | A square root of a nonnegative number n is a nubmer r such that $r^2 = n$. eg0: All positive numbers have two square roots, one positive and one negative. eg1: The only square root of 0 is 0. eg2: The expression consisting of the square root symbol $\sqrt \ $ placed over a nonnegative number denotes the nonnegative square root. $\sqrt{100} = 10$, while $\pm 10$ are square roots of 100. |
cube root | 立方根 | There are some notable differences between odd order roots and even order roots (in real number system): For odd order roots, there is exactly one root for every number n, even when n is negative. For even order roots, there are exactly two roots for every positive number n and no roots for any negative number n. |
fourth root | 四次方根 |
English | Chinese | Remark & Example |
---|---|---|
decimal | 小数 | 7532.418 $\Rightarrow$ [Thousands/Hundreds/Tens/Ones or Units/Tenths/Hundredths/Thousandths] |
terminate | v. 有限小数（有理数） | The decimal that results from the long division will either terminate, as in $\frac 1 4 = 0.25$, or repeat without end, as in $\frac 1 9 = 0.111…$. |
repeat | v. 无限循环小数（有理数） | One way to indicate the repeating part of a decimal that repeats without end is to use a bar over the digits that repeat. eg: $\frac {15}{14} = 1.0\overline{714285}$ |
irrational numbers | 无理数 | fraction with integers in the numerator and denominator $\Longleftrightarrow$ rational number $\Longleftrightarrow$ terminating or repeating decimal |
English | Chinese | Remark & Example |
---|---|---|
real numbers | 实数 | The set of real numbers consists of all rational numbers and all irrational numbers. The real numbers include all integers, fractions, and decimals. |
real number line | 实数线 | The set of real numbers can be represented by a number line called the real number line. |
less than or equal to | 小于等于 | |
greater than or equal to | 大于等于 | |
interval | 区间 | |
endpoint | 区间端点 | |
absolute value | 绝对值 | |
triangle inequality | 三角不等式 | $|r + s| \le |r| + |s|$ |
English | Chinese | Remark & Example |
---|---|---|
ratio | 比 | eg0: s/t, “s to t”, s:t eg1: “r to s to t” |
lowest terms | 最简分数/既约分数 | Like fractions, ratios can be reduced to lowest terms. |
proportion | 比例式 | A proportion is an equation relating two ratios. eg: $\frac{9}{12} = \frac{3}{4}$ |
cross multiplication | 交叉相乘 | To solve a problem involving ratios, you can often write a proportion and solve it by cross multiplication. eg: 9*4=12*3 |
English | Chinese | Remark & Example |
---|---|---|
percent | 百分比 | The term percent means per hundred, or hundredths. |
base | （百分比里的）分母 | |
percent change | 百分比变化量（增长率，衰减率） | When a quantity changes from an initial positive amount to another positive amount, you can compute the amount of change as a percent of the initial amount. This is called percent change. percent increase, percent decrease |
The review of algebra begins with algebraic expressions, equations, inequalities, and functions and then progresses to several examples of applying them to solve real-life word problems. The review of algebra ends with coordinate geometry and graphs of functions as other important algebraic tools for solving problems.
代数：代数表达式，等式/方程，不等式，函数，代数应用，坐标几何，函数图像，代数解题工具
English | Chinese | Remark & Example |
---|---|---|
variable | 变量 | A variable is a letter that represents a quantity whose value is unknown. |
algebra expression | 代数表达式 | An algebraic expression has one or more variables and can be written as a single term or as a sum of terms. eg: $w^3z + 5z^2 - z^2 + 6$ has four terms, and $\frac{8}{n + p}$ has one term. |
term | 式子/（代数式的）项 | |
like term | 同类项（次数相同） | Like terms have the same variables, and the corresponding variables have the same exponents. |
constant term | 常数项 | A term that has no variable is called a constant term. |
coefficient | 项的系数 | A number that is multiplied by variables is called the coefficient of a term. eg: Coefficient of 5/y is 5. |
polynomial | 多项式 | A polynomial is the sum of a finite number of terms in which each term is either a constant term or a product of a coefficient and one or more variables with positive integer exponents. eg: The expression $4x^6 + 7x^5 - 3x + 2$ is a polynomial in one variable, x. The polynomial has four terms. |
degree | 度/次数 | The degree of each term is the sum of the exponents of the variables in the term. The degree of a polynomial is the greatest degree of its terms. eg: Polynomials of degrees 2 and 3 are known as quadratic and cubic polynomials, respectively. |
identity | 相等 | A statement of equality between two algebraic expressions that is true for all possible values of the variables involved is called an identity. eg: $(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3$ |
linear equation | 线性方程 | eg: 3z + 5 - t = 20 is a linear equation in two variables. |
quadratic equation | 二次方程 | eg: 20 x^2 = 10 is a quadratic equation in one variables. |
quadratic, cube, biquadratic/quadruplicate/biquadrate | 二次方，三次方，四乘幂/四倍/双二次 |
English | Chinese | Remark & Example |
---|---|---|
base | 基数 | In the algebraic expression $x^a$, where x is raised to the power a, x is called the base and a is called the exponent. |
exponent | 指数 |
English | Chinese | Remark & Example |
---|---|---|
equation | 方程 | An equation is a statement of equality between two mathematical expressions. |
solution | 方程的解 | The values of the variables that make the equation true are called the solutions of the equation. |
solve an equation | 解方程 | To solve an equation means to find the values of the variables that make the equation true, that is, the values that satisfy the equation. Two equations that have the same solutions are called equivalent equations. |
satisfy the equation | 满足方程 | |
equivalent equation | 具有相同解的方程 | The general method for solving an equation is to find successively simpler equivalent equations so that the simplest equivalent equation makes the solutions obvious. eg: $x + 1 = 2$ and $2x + 2 = 4$ are equivalent equations. |
linear equation | 线性方程 | A linear equation is an equation involving one or more variables in which each terms in the equation is either a constant term or a variable multiplied by a coefficient. None of the variables are multiplied together or raised to a power greater than 1. It is possible for a linear equation to have no solutions. Also, it is possible that what looks to be a linear equation could turn out to be an identity when you try to solve it. |
ordered pair | 有序数对 | A solution of linear equation in two variables is an ordered pair of numbers (x, y) that makes the equation true when the values of x and y substituted into the equation. |
system of equations | 方程组 | A set of equations in two or more variables is called a system of equations. There are two basic methods for solving systems of linear equations, by substitution of by elimination. |
simultaneous equations | 联立方程（方程组里的方程互称） | The equations in the system are called simultaneous equations. |
substitution | 带入消元解法 | In the substitution method, one equation is manipulated to express one variable in terms of the other. Then the expression is substituted in the other equation. |
elimination | 加减消元解法 | In the elimination method, the object is to make the coefficients of one variable the same in both equations so that one variable can be eliminated either by adding the equations together or by subtracting one from the other. |
English | Chinese | Remark & Example |
---|---|---|
quadratic equation | 二次方程 | A quadratic equation in the variable x is an equation that can be written in the form $ax^2 + bx + c = 0$, where a, b, c are real numbers and a $\not =$ 0. Quadratic equations have zero, one or two real solutions. |
quadratic formula | 二次方程求根公式 | One way to find solutions of a quadratic equation is to use the quadratic formula: $\displaystyle x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ |
English | Chinese | Remark & Example |
---|---|---|
inequality | 不等式 | |
solve an inequality | 解不等式 | To solve an inequality means to find the set of all values of the variable that make the inequality true. This set of values is also known as the solution set of an inequality. Two inequalities that have the same solution set are called equivalent inequalities. |
solution set | 不等式解集 | |
equivalent inequalities | 等价的不等式 |
English | Chinese | Remark & Example |
---|---|---|
function | 函数 | eg: f(x) is called the value of f at x. |
domain | 自变量的取值范围（定义域） | The domain of a function is the set of all permissible inputs, that is, all permissible values of the variable x. Without an explicit restriction, the domain is assumed to be the set of all values of x for which f(x) is a real number. |
Translating verbal descriptions into algebraic expressions is an essential initial step in solving word problems.
English | Chinese | Remark & Example |
---|---|---|
product | 乘积 | |
interest | 利息 | |
simple interest | 单利 | Simple interest is based only on the initial deposit, which serves as the amount on which interest is computed, called the princial, for the entire time period. eg: If the amount P is invested at a simple annual interest rate of r percent, then the value V of the investment at the end of t years is given by the formula $\displaystyle V = P (1 + \frac {rt}{100})$. |
principal | 本金 | |
compound interest | 复利 | In the case of compound interest, interest is added to the principal at regular time intervals, such as annually, quarterly, and monthly. Each time interest is added to the principal, the interest is said to be compounded. After each compounding, interest is earned on the new principal, which is the sum of the preceding principal and the interest just added. eg0: If the amount P is invested at an annual interest rate of r percent, compounded annually, then the value V of the investment at the end of t years is given by the formula $\displaystyle V = P (1 + \frac{r}{100})^{t}$. eg1: If the amount P is invested at an annual interest rate of r percent, compounded n times per year, then the value V of the investment at the end of t years is given by the formula $\displaystyle V = P (1 + \frac{r}{100})^{nt}$. |
English | Chinese | Remark & Example |
---|---|---|
coordinate | 坐标 | |
rectangular coordinate system / xy-coordinate system / xy-plane | 直角坐标系 | Two real number lines that are perpendicular to each other and that intersect at their respective zero points define a rectangular coordinate system, often called the xy-coordinate system or xy-plane. The horizontal number line is called the x-axis and the vertical number line is called the y-axis. The point where the two axes intersect is called the origin, denoted by O. The two axes divide the plane into four regions called quadrants. |
x-axis y-axis | x轴 y轴 | a and b are symmetric about the x-axis = a is the reflection of b about the x-axis 关于x轴对称 |
origin | 原点O | about the origin 关于原点 |
quadrant | 象限 | |
x-coordinate y-coordinate | 横坐标 纵坐标 | Each point J in the xy-plane can be identified with an ordered pair (x, y) of real numbers and is denoted by J (x, y). The first number in the ordered pair is called the x-coordinate, and the second number is called the y-coordinate. |
graph of an equation | 方程的图形表示 | The graph of an equation in the variables x and y is the set of all points whose ordered pairs (x, y) satisfy the equation. |
line of symmetry | 对称轴 | y = x is a line of symmetry for the graphs of y = 5x + 6 and x = 5y + 6 |
slope | 斜率 | $\displaystyle \frac{y_2 - y _1}{x_2 - x_1}$, this ratio is often called “rise over run”, where rise is the change in y and run is the change in x. |
y-intercept x-intercept | 纵截距 横截距 | Sometimes the terms x-intercept and y-intercept refer to the actual intersection points. |
parallel | 平行 | Two lines are parallel if their slopes are equal. |
perpendicular | 垂直 | Two lines are perpendicular if their slopes are negative reciprocals of each other. |
intersect | 相交 | |
parabola | 抛物线 | The graph of a quadratic equation of the form $y = ax^2 + bx + c$, where a, b and c are constants and $a \not = 0$, is a parabola. |
vertex | 抛物线顶点 | If a is negative, the parabola opens upward and the vertex is its lowest point. |
circle | 圆 | The graph of an equation of the form $(x - a)^2 + (y - b)^2 = r^2$ is a circle with its center at the point (a, b) and with radius r > 0. |
Notice the fine distinction between the graphs of equations and ones of functions. To graph a function in the xy-plane, you represent each input x and its corresponding output f(x) as a point (x, y), where y = f(x). In other words, you use the x-axis for the input and the y-axis for the output.
English | Chinese | Remark & Example |
---|---|---|
piecewise-defined function | 分段函数 | |
dashed curve | 虚线 | |
reflection | 对称 | |
shifted upward / downward / to the left / to the right | 平移 | |
stretched / dilated / shrunk / contracted | 拉伸/扩大/压缩/缩小 | eg: The graph of ch(x) is the graph of h(x) stretched vertically by a factor of c if c > 1. |
The review of geometry begins with lines and angles and progresses to other plane figures, such as polygons, triangles, quadrilaterals, and circles. The review of geometry ends with some basic three-dimensional figures. Coordinate geometry is covered in the Algebra part.
几何：线，角，平面图形（多边形，三角形，四边形，圆），基本三维图形，坐标几何（代数部分已提及）
English | Chinese | Remark & Example |
---|---|---|
line | 直线 | A line is understood to be a straight line that extends in both directions without ending. |
line segment | 线段 | Given any two points on a line, a line segment is the part of the line that contains the two points and all points between them. The two points are called endpoints. |
endpoints | 端点 | |
congruent line segments | 等长线段 | Line segments that have equal lengths are called congruent line segments. |
midpoint | 中点 | The point that divides a line segment into two congruent line segments is called the midpoint of the line segment. |
length | 长度 | Sometimes the notation AB denotes line segment AB, and sometimes it denotes the length of line segment AB. The meaning of the notation can be determined from the context. |
angles | 角 | When two lines intersect at a point, they form four angles. Each angle has a vertex at the point of intersection of the two lines. Somtimes the angle symbol $\angle$ is used instead of the word “angle.” |
vertex | 顶点 | |
opposite angles, vertical angles | 对顶角 | |
congruent angles | 等角 | Opposite angles have equal measure, and angles that have equal measure are called congruent angles. Hence, opposite angles are congruent. |
perpendicular lines | 垂线 | Two lines that intersect to form four congruent angles are called perpendicular lines, for example k $\perp$ m. Each of the four angles has a measure of 90$^{\circ}$. |
right angle | 直角 | An angle with a measure of 90$^{\circ}$ is called a right angle. A common way to indicate that an angle is a right angle is to draw a small square at the vertex of the angle. |
acute angle obtuse angle | 锐角 钝角 | An angle with measure less than 90$^{\circ}$ is called an acute angle, and an angle with measure between 90$^{\circ}$ and 180$^{\circ}$ is called an obtuse angle. |
parallel lines | 平行线 | Two lines in the same plan that do not intersect are called parallel lines, for example k $\parallel$ m. |
English | Chinese | Remark & Example |
---|---|---|
polygon | 多边形 | A polygon is a closed figure formed by three or more line segments all of which are in the same plane. The line segments are called the sides of the polygon. Each side is joined to two other sides at its endpoints, and the endpoints are called vertices (pl. plural of vertex). If a polygon has n sides, it can be divided into n - 2 triangles. Since the sum of measures of the interior angles of a triangle is 180$^{\circ}$, it follows that the sum of the measures of the interior angles of an n-sided polygon is $(n - 2)(180^{\circ})$. |
side | 边 | |
vertices | 顶点 | |
convex | 凸面的 | convex polygon 凸多边形 |
interior angle exterior angle | 内角 外角 | |
diagonal | 对角线 | |
adjacent | 相邻的 | nonadjacent vertices 非相邻顶点 |
triangle | 三角形 | |
quadrilateral | 四边形 | |
pentagon | 五角形 | |
hexagon | 六角形 | |
octagon | 八角形 | |
regular polygon | 正多边形 | eg: In a regular octagon the measure of each angle is $\frac {1080^{\circ}}{8} = 135^{\circ}$. |
perimeter | 周长 | The perimeter of a polygon is the sum of the lengths of its sides. |
area | 面积 | The area of a polygon refers to the area of the region enclosed by the polygon. |
radius diameter | 半径 直径 |
Every triangle has three sides and three interior angles.
The measures of the interior angles add up to $180^{\circ}$.
The length of each side must be less than the sum of the lengths of the other two sides.
English | Chinese | Remark & Example |
---|---|---|
equilateral triangle | 等边三角形 | A triangle with three congruent sides is called an equilateral triangle. |
isosceles triangle | 等腰三角形 | A triangle with at least two congruent sides is called an isosceles triangle. |
right triangle | 直角三角形 | A triangle with an interior right angle is called a right triangle. The side opposite the right angle is called the hypotenuse; the other two sides are called legs. |
hypotenuse legs | （直角三角形的）斜边，弦 （直角三角形的腰 | |
congruent triangles | 全等三角形 | |
similar triangles | 相似三角形 | |
included angle | 夹角 | |
scale factor of similarity | 相似尺度因子，比例因子 |
English | Chinese | Remark & Example |
---|---|---|
rectangle | 矩形 | A quadrilateral with four right angles is called a rectangle. Opposite sides of a rectangle are parallel and congruent, and the two diagonals are also congruent. |
square | 正方形 | A rectangle with four congruent sides is called a square. |
parallelogram | 平行四边形 | A quadrilateral in which both pairs of opposite sides are parallel is called a parallelogram. In a parallelogram, opposite sides are congruent and opposite angles are congruent. |
trapezoid | 梯形 | A quadrilateral in which at least one pair of opposite sides is parallel is called a trapezoid. Two opposite, parallel sides of the trapezoid are called bases of the trapezoid. |
English | Chinese | Remark & Example |
---|---|---|
circle | 圆 | Given a point O in a plane and a positive number r, the set of points in the plane that are a distance of r units from O is called a circle. The point O is called the center of the circle and the distance r is called the radius of the circle. The diameter of the circle is twice the radius. |
center | 圆心 | |
radius | 半径 | |
diameter | 直径 | |
congruent circles | 等圆 | Two circles with equal radius are called congruent circles. |
chord | 弦 | Any line segment joining two points on the circle is called a chord. The terms “radius” and “diameter” can also refer to line segments: A radius is any line segment joining a point on the circle and the center of the circle, and a diameter is a chord that passes through the center of the circle. |
circumference | 圆周 | The distance around a circle is called the circumference of the circle, which is analogous to the perimeter of a polygon. The ratio of the circumference C to the diameter d is the same for all circles and is denoted by the Greek letter $\pi$, that is, $\frac{C}{d} = \pi$, the value of $\pi$ is approximately 3.14 and can also be approximated by the fraction $\frac{22}{7}$. $C = \pi d = 2\pi r$ |
arc | 弧 | Given any two points on a circle, an arc is the part of the circle containing the two points and all the points between them. Two points on a circle are always the endpoints of two arcs. An arc is frequently identified by three points to avoid ambiguity. |
central angle | 圆心角 | A central angle of a circle is an angle with its vertex at the center of the circle. |
measure of an arc | 弧的量度 | The measure of an arc is the measure of its central angle, which is the angle formed by two radius that connect the center of the circle to the two endpoints of the arc. |
length of an arc | 弧的长度 | The ratio of the length of an arc to the circumference is equal to the ratio of the degree measure of the arc to 360$^{\circ}$. |
area | 面积 | The area of a circle with radius r is equal to $\pi r^2$. |
sector | 扇形 | A sector of a circle is a region bounded by an arc of the circle and two radius. |
area of a sector | 扇形面积 | To find the area of a sector, note that the ratio of the area of a sector of a circle to the area of the entire circle is equal to the ratio of the degree measure of its arc to 360$^{\circ}$. |
tangent | 正切，切线 | A tangent to a circle is a line that lies in the same plane as the circle and intersects the circle at exactly one point, called the point of tangency. If aline is tangent to a circle, then a radius drawn to the point of tangency is perpendicular to the tangent line. |
point of tangency | 切点 | |
inscribed circumscribed | 内接的，内切的/雕刻，题写 外接的，外切的/局限，限定 | A polygon is inscribed in a circle if all its vertices lie on the circle, or equivalently, the circle is circumscribed about the polygon. 内接多边形，外接圆 eg0: It is not always the case that if a triangle is inscribed in a circle, the center of the circle is inside the inscribed triangle. It is also possible for the center of the circle to be outside the inscribed triangle, or on one of the sides of inscribed triangle. eg1: If one side of an inscribed triangle is a diameter of the circle, then the triangle is a right triangle. A polygon is circumscribed about a circle if each side of the polygon is tangent to the circle, or equivalently, the circle is inscribed in the polygon. 外切多边形，内切圆 |
concentric circles | 同心圆 | Two or more circles with the same center are called concentric circles. |
Basic three-dimensional figures include rectangular solids, cubes, cylinders, spheres, pyramids, and cones. In this section, we look at some properties of rectangular solids and right circular cylinders.
English | Chinese | Remark & Example |
---|---|---|
solids, cubes, cylinders, spheres, pyramids, cones, cuboids | 固体，立方体，圆柱体，球体，金字塔/角锥，圆锥，长方体 | |
rectangular solid, rectangular prism | 长方体，矩形棱镜，直角棱镜 | A rectangular solid, or rectangular prism, has 6 rectangular surfaces called faces. Adjacent faces are perpendicular to each other. Each line segment that is the intersection of two faces is called an edge, and each point at which the edges intersect is called a vertex. There are 12 edges and 8 vertices. The dimensions of a rectangular solid are the length $\ell$, the width $w$, and the height $h$. eg: A rectangular solid with six square faces is called a cube, in which case $\ell = w = h$. |
faces | 面 | |
edge | 边 | |
vertex | 顶点 | |
volume | 体积/音量 | The volume V of a rectangular solid is the product of its three dimensions, or $V = \ell w h$. |
surface area | 表面积 | The surface area A of a rectangular solid is the sum of the areas of the six faces, or $A = 2(\ell w + \ell h + wh)$. |
circular cylinder | 圆柱体 | A circular cylinder consists of two bases that are congruent circles lying in parallel planes and a lateral surface made of all line segments that join points on the two circles and that are parallel to the line segment joining the centers of the two circles. The latter line segment is called the axis of the cylinder. |
lateral surface | 侧面 | |
axis | 轴线 | |
right circular cylinder | 正圆柱体 | A right circular cylinder is a circular cylinder whose axis is perpendicular to its bases. The height of a right circular cylinder is the perpendicular distance between the two bases, which is equal to the length of the axis. $V = \pi r^2 h, \ \ A = 2(\pi r^2) + 2\pi r h$ |
height | 高 |
The review of data analysis begins with methods for presenting data, followed by counting methods and probability, and then progresses to distributions of data, random variables, and probability distributions. The review of data analysis ends with examples of data interpretation.
数据分析：数据表示，数数，概率，数据分布，随机变量，概率分布，数据演绎
English | Chinese | Remark & Example |
---|---|---|
quantitative, numerical | 定量的，数字的 | |
categorical, nonnumerical | 分类的，非数字的 | |
distribution of a variable, distribution of data | 变量的分布，数据的分布 | Data are collected from a population by observing one or more variables. The distribution of a variable, or distribution of data, indicates how frequently different categorical or numerical data values are observed in the data. |
frequency, count | 频率（频数） | The frequency, or count, of a particular category or numerical value is the number of times that the category or numerical value appears in the data. |
frequency distribution | 频率（频数）分布 | A frequency distribution is a table or graph that presents the categories or numerical values along with their corresponding frequencies. |
relative frequency | 相对频率（频率） | The relative frequency of a category or a numerical value is the corresponding frequency divided by the total number of data. Relative frequencies may be expressed in terms of percents, fractions, or decimals. |
relative frequency distribution | 相对频率（频率）分布 | A relative frequency distribution is a table or graph that presents the relative frequencies of the categories or numerical values. |
bar graph, bar chart | 条形图 | In a bar graph, each of the data categories or numerical values is represented by a rectangular bar, and the height of each bar is proportional to the corresponding frequency or relative frequency. All of the bars are drawn with the same width, and the bars can be presented either vertically or horizontally. |
segmented bar graph, stacked bar graph | 分段条形图，堆积条形图 | A segmented bar graph, or stacked bar graph, is similar to a regular bar graph except that in a segmented bar graph, each rectangular bar is divided, or segmented, into smaller rectangles that show how the variables is “separated” into other related variables. |
histogram | 直方图，柱状图 | Histograms are graphs of frequency distributions that are similar to bar graphs, but they must have a number line for the horizontal axis, which represents the numerical variable. Also, in a histogram, there are no regular spaces between the bars. Any spaces between bars in a histogram indicate that there are no data in the intervals represented by the spaces. Histograms are useful for identifying the general shape of a distribution of data. |
circle graphs, pie charts | 饼图 | Circle graphs, often called pie charts, are used to represent data that have been separated into a small number of categories. They illustrate how a whole is separated into parts. |
sector | 扇形，扇区 | Each part of a circle graph is called a sector. |
scatterplot | 散点图 | A scatterplot is a type of graph that is useful for showing the relationship between two numerical variables whose values can be observed in a single population of individuals or objects. |
trend | 趋势，走向，倾向 | A scatterplot makes it possible to observe an overall pattern, or trend, in the relationship between the two variables. Also, the strength of the trend as well as striking deviations from the trend are evident. The trend line can be used to make predictions. |
line graph | 线图 | A line graph is another type of graph that is useful for showing the relationship between two numerical variables, especially if one of the variables is time. There is at most one data point for each value on the horizontal axis, similar to a function. The data points are in order from left to right, and consecutive data points are connected by a line segment. |
time series | 时间序列 | When one of the variables is time, it is associated with the horizontal axis, which is labeled with regular time intervals. |
English | Chinese | Remark & Example |
---|---|---|
statistics, statistical measures | 统计量，统计度量 | Data can be described numerically by various statistics, or statistical measures. These statistical measures are often grouped in three categories: measures of central tendency, measures of position, and measures of dispersion. |
central tendency | 集中趋势（中心趋势，指均数、中数、众数） | Measures of central tendency indicate the “center” of the data along the number line and are usually reported as values that represent the data. There are three common measures of central tendency: 1. the arithmetic mean/average/mean; 2. the median; 3. the mode. |
arithmetica mean / average / mean | 算术平均数 | |
median | 中位数 | The median is a measure of central tendency that is fairly unaffected by unusually high or low values relative to the rest of the data. |
mode | 众数 | The mode of a list of numbers is the number that occurs most frequently in the list. |
weighted mean | 加权平均数 | |
weight | 权重 | |
position | 位置，分位 | The three most basic positions, or locations, in a list of numerical data ordered from least to greatest are the beginning, the end, and the middle. It is useful here to label these as L for the least, G for the greatest, and M for the median. Aside from these, the most common measures of position are quartiles and percentiles. As with the mean and the median, the quartiles and percentiles may or may not themselves be values in the data. |
quartiles | 四分位数 | There are three quartile numbers, called the first quartile ($Q_1$) , the second quartile ($Q_2$) , and the third quartile ($Q_3$) , that divide the data into four roughly equal groups. There are various rules to determine the exact values of $Q_1$ and $Q_3$, and the most common rule is that $Q_1$ is the median of the first half of the data in the ordered list and $Q_3$ is the median of the second half of the data in the ordered list. eg: 4 is in the first quartile. The phase “in a quartile” refers to being in one of the four groups determined by $Q_1$, $Q_2$, and $Q_3$. |
percentiles | 百分位数 | There are 99 percentile numbers that divide the data into 100 roughly equal groups ($P_1, P_2, P_3, … , P_{99}$) . Consequently, $Q_1 = P_{25}, M = Q_2 = P_{50}, Q_3 = P_{75}$. |
dispersion | 散布，离差，差量 | Measures of dispersion indicate the degree of spread of the data. Tho most common statistics used as measures of dispersion are the range, the interquartile range, and the standart deviation. |
range | 极差 | The range of the numbers in a group of data is the difference between the greatest number G in the data and the least number L in the data; that is, G - L. |
outlier | 离群值，异常值 | Outliers are unusually small or unusually large in comparison with the rest of the data. |
interquartile | 四分位数，四分点 | |
interquartile range | 四分位数间距，四分位差 | A measure of dispersion that is not usually affected by outliers is the interquartile range. It is defined as the difference between the third quartile and the first quartile; that is, $Q_3 - Q_1$. |
boxplot， box-and-whisker plot | 箱线图，箱形图，盒须图 | $L, Q_1, Q_2, Q_3, G$ can be plotted along a number line to show where the four quartile groups lie. Such plots are called boxplots or box-and-whisker plots, because a box is used to identify each of the two middle quartile groups of data, and “whiskers” extend outward from the boxes to the least and greatest values. |
standard deviation (population standard deviation) | 标准差 | The standard deviation is a measure of spread that depends on each number in the list. Using the mean as the center of the data, the standard deviation takes into account how much each value differs from the eman and then takes a type of average of these differences. |
sample standard deviation | 样本标准差 | The sample standard deviation is qualified with the word “sample” and is computed by dividing the sum of the squared differences by n - 1 instead of n. |
standardization | 标准化 | The process of subtracting the mean from each value and then dividing the result by the standard deviation is called standardization. Standardization is a useful tool because for each data value, it provides a measure of position relative to the rest of the data independently of the variable for which the data was collected and the units of the variable. (?) |
Measures of Central Tendency: P140
Measures of Position: P142
Measures of Dispersion: P143
Example 4.2.9: P146 (Calculating standard deviations)
Fact about standard deviation: In any group of data, most of the data are within 3 standard deviations of the mean.
Thus, when any group of data are standardized, most of the data are transformed to an interval on the number line centered about 0 and extending from -3 to 3. The mean is always transformed to 0.
English | Chinese | Remark & Example |
---|---|---|
set | 集合 | The term set has been used informally in this review to mean a collection of objects that have some property. |
members, elements | 元素 | The objects of a set are called members or elements. |
finite | 有限的（集合） | Some sets are finite, which means that their members can be completely counted. eg: For any finite set S, the number of elements of S is denoted by |S|. |
infinite | 无限的（集合） | Sets that are not finite are called infinite sets, such as the set of all integers. |
empty set | 空集 | A set that has no members is called the empty set and is denoted by the symbol $\varnothing$. |
nonempty | 非空集 | A set with one or more members is called nonempty. |
subset | 子集 | If A and B are sets and all of the members of A are also members of B, then A is a subset of B. |
list | 表 (?) | A list is like a finite set, having members that can all be listed, but with two differences. In a list, the members are ordered-that is, rearranging the members of a list makes it a different list. Thus, the terms “first element,”“second element,” and so on, make sense in a list. Also, elements can be repeated in a list and the repetitions matter. |
intersection | 交集 | The intersection of S and T is the set of all elements that are in both S and T and is denoted by $S \cap T$. |
union | 并集 | The union of S and T is the set of all elements that are in either S or T or both and is denoted by $S \cup T$. |
disjoint / mutually exclusive | 互斥，不相交 | If sets S and T have no elements in common, they are called disjoint or mutually exclusive. |
Venn diagram | 文氏图 | In a Venn diagram, sets are represented by circular regions that overlap if they have elements in common but do not overlap if they are disjoint. |
universal set | 全集 | Sometimes the circular regions are drawn inside a rectangular region, which represents a universal set, of which all other sets involved are subsets. |
inclusion-exclusion principle | 容斥原理 | The number of elements in the union of two sets equals the sum of their individual numbers of elements minus the number of elements in their intersection. | A $\cup$ B | = | A | + | B | - | A $\cap $ B | |
multiplication principle | 乘法原理 | Suppose there are two choices to be made sequentially and that the second choice is independent of the first choice. Suppose also that there are k different possibilities for the first choice and m different possibilities for the second choice. The multiplication principle states that under those conditions, there are km different possibilities for the pair of choices. |
permutation | 排列 | The number of ways to order the n objects is equal to the product $n(n-1)(n-2)…(3)(2)(1)$. Each order is called a permutation, and the product above is called the number of permutations of n objects. |
factorial | 阶乘 | Because products of the form $n(n-1)(n-2)…(3)(2)(1)$ occur frequently when counting objects, a special symbol $n!$, called n-factorial, is used to denote this product. |
permutations of n objects taken k at a time | n中选k的排列数 | $\displaystyle P_n^k =\ _nP_k = \frac{n!}{(n-k)!}$ |
combination | 组合 | selecting without order $(\text{number of ways to select without order}) = \frac{\text{number of ways to select wit order}}{\text{(number of ways to order)}}$ |
combinations of n objects taken k at a time / n choose k | n中选k的组合数 | $\displaystyle C_n^k =\ _nC_k = \binom{n}{k} = \frac{n!}{k!(n-k)!}$ eg0: n choose 0 is 1 eg1: n choose n is 1 eg2: n choose k = n choose (n-k) |
English | Chinese | Remark & Example |
---|---|---|
probability experiment, random experiment | 概率实验，随机实验 | A probability experiment, also called a random experiment, is an experiment for which the result, or outcome, is uncertain. We assume that all of the possible outcomes of an experiment are known before the experiment is performed, but which outcome will actually occur is unknown. |
outcome | 结果，输出，取值 | |
sample space | 样本空间 | The set of all possible outcomes of a random experiment is called the sample space, and any particular set of outcomes is called an event. |
event | 事件 | eg0: The event that both E and F occur, that is, outcomes in the set $E \cap F$. eg1: The event that E or F, or both, occur, that is, outcomes in the set $E \cup F$. eg2: It is common to use the shorter notation “E and F” instead of “both E and F occur” and use “E or F” instead of “E or F or both occur.” |
probability | 概率 | The probability of an event is a number from 0 to 1, inclusive, that indicates the likelihood that the event occurs when the experiment is performed. eg0: If an event E is certain to occur, then P(E) = 1. eg1: If an event E is certain not to occur, then P(E) = 0. eg2: If an event E is possible but not certain to occur, then 0 < P(E) < 1. eg3: The probability that an event E will not occur is equal to 1 - P(E). eg4: If E is an event, then the probability of E is the sum of the probabilities of the outcomes in E. eg5: The sum of the probabilities of all possible outcomes of an experiment is 1. eg6: P(either E or F, or both, occur) = P(E) + P(F) - P(both E and F occur), which is the inclusion-exclusion principle applied to probability. eg7: If E and F are mutually exclusive, then P(both E and F occur) = 0, and therefore, P(either E of F, or both, occur) = P(E) + P(F). eg8: E and F are said to be independent if the occurrence of either event does not affect the occurrence of the other. If two events E and F are independent, then P(both E and F occur) = P(E)P(F). eg9: Note that if P(E) $\not =$ 0 and P(F) $\not =$ 0, then events E and F cannot be both mutually exclusive and independent. For if E and F are independent, then P(both E and F occur) = P(E)P(F) $\not =$ 0, but if E and F are mutually exclusive, then P(both E and F occur) = 0. |
random selection | 随机选择，随机抽样 | The assumption of random selection means that each of the names is equally likely to be selected. |
equally likely | 等可能，等概率 | In general, for a random experiment with a finite number of possible outcomes, if each outcome is equally likely to occur, then the probability that an event E occurs is defined by $\displaystyle P(E) = \frac{\text{the number of outcomes in the event E}}{\text{the number of possible outcomes in the experiment}}$ |
mutually exclusive | 互斥 | Events that cannot occur at the same time are said to be mutually exclusive. |
independent | 独立 | (Notice the difference between “mutually exclusive” and “independent”. “Mutually exclusive” means there is only one experiment with many outcomes. “Independent” means two experiments does not affect each other.) |
In data analysis, variables whose values depend on chance play an important role in linking distributions of data to probability distributions. Such variables are called random variables.
English | Chinese | Remark & Example |
---|---|---|
class | 类 | eg: The measurements were grouped into 50 intervals, or classes, of 10 hours each. |
distribution curve, density curve, frequency curve | 分布曲线，密度曲线，频率曲线 | The distribution can be modeled by a smooth curve that is close to the tops of the bars. Such a model retains the shape of the distribution but is independent of classes. The area under the curve that models the distribution is 1. This model curve is called a distribution curve, but it has other names as well, including density curve and frequency curve. The purpose of the distribution curve is to give a good illustration of a large distribution of numerical data that does not depend on specific classes. |
random variable | 随机变量 | Given a distribution of data, a variable, say X, may be used to represent a randomly chosen value from the distribution. Such a variable X is an example of a random variable, which is a variable whose value is a numerical outcome of a random experiment. The concept of a random variable is more general than representing a randomly chosen value from a distribution of data. A random variable can be any quantity whose value is the result of a random experiment. The possible values of the random variable are the same as the outcomes of the experiment. So any random experiment with numerical outcomes naturally has a random variable associated with it. |
probability distribution | 概率分布 | A fundamental link between data distributions and probability distributions: For a random variable that represents a randomly chosen value from a distribution of data, the probability distribution of the random variable is the same as the relative frequency distribution of the data. |
mean of the random variable X / expected value | 随机变量X的均值/数学期望，预期值 | Another name for the mean of a random variable is expected value. The mean of the random variable X is the sum of the products XP(X) for all values of X, that is, the sum of each value of X multiplied by its corresponding probability P(X). |
discrete random variables | 离散随机变量 | Whose values consist of discrete points on a number line. Fundamental Link: In a histogram representing the probability distribution of a random variable, the area of each bar is proportional to the probability represented by the bar. |
uniform distribution | 均匀分布 | Each of the bars in the histogram of the probability distribution would have the same height. Such a flat histogram indicates a uniform distribution, since the probability is distributed uniformly over all possible outcomes. |
approximately normally distributed | 近似正态分布 | Many natural processes yield data that have a relative frequency distribution shaped somewhat like a bell. Such data are said to be approximately normally distributed and have four properties. Property 1: The mean, median, and mode are all nearly equal. Property 2: The data are grouped fairly symmetrically about the mean. Property 3: About two-thirds of the data are within 1 standard deviation of the mean. Property 4: Almost all of the data within 2 standard deviations of the mean. |
continuous probability distribution | 连续概率分布 | The region below a distribution curve represents a distribution called a continuous probability distribution. There are many different continuous probability distributions, but the most important one is the normal distribution, which has a bell-shaped curve. The area of the region under the curve is 1, and the areas of vertical slices of the region, like the areas of the bars of a histogram, are equal to probabilities of a random variable associated with the distribution. Such a random variable is called a continuous random variable. |
normal distribution | 正态分布 | The properties listed above for the approximately normal distribution of data hold for the normal distribution, except that the mean, median, and mode are exactly the same and the distribution is perfectly symmetric about the mean. A normal distribution, though always shaped like a bell, can be centered around any mean and can be spread out to a greater or lesser degree, depending on the standard deviation. The less the standard deviation, the less spread out the curve is; that is to say, at the mean the curve is higher and as you move away from the mean in either direction it drops down toward the horizontal axis faster. |
continuous random variable | 连续随机变量 | Continuous random variable plays the same role as a random variable that represents a randomly chosen value from a distribution of data. The main difference is that we seldom consider the event in which a continuous random variables is equal to a single values like X=3; rather, we consider events that are described by intervals of values such as 1 < X < 3 and X > 10. |
standard normal distribution | 标准正态分布 | The standard normal distribution is a normal distribution with a mean of 0 and standard deviation equal to 1. To transform a normal distribution with a mean of m and a standard deviation of d to a standard normal distribution, you standardize the values; that is, you subtract m from any observed value of the normal distribution and then divide the result by d. eg0: 1 deviation from the mean: P = 0.683 eg1: 3 deviation from the mean: P = 0.9987 |
算术：整数，分数，小数，实数集，加减乘除，指数和开方，比例和百分数
代数：代数表达式，等式/方程，不等式，函数，代数应用，坐标几何，函数图像，代数解题工具
几何：线，角，平面图形（多边形，三角形，四边形，圆），基本三维图形，坐标几何（代数部分已提及）
数据分析：数据表示，数数，概率，数据分布，随机变量，概率分布，数据演绎
Appendix 0. 多边形
Triangle 三角形
rectangle 矩形
square 正方形
quadrilateral/quadrangle/tetragon 四边形 (later是边，angle是角)
pentagon 五边形
hexagon 六边形
septangle/heptagon/sepilateral 七边形
octagon 八边形
decagon 十边形
hexadecagon 十六边形
Appendix 1. 英语数字前缀
English | Latin | Greek | Example |
---|---|---|---|
one | uni | mono | unique, uniform, unipolar(单极的); monologue(独白), monogamy, monochrome, monopolize(垄断，独占), monotone(单调音，单色调) |
two | du/bi | dis/dy/di | bilingual(双语的), bilateral(双边的), bimonthly; dioxide, disyllable(双音节词), digraph(双字一音；有向图) |
three | tri | tri | triangle, tricycle, trilogy(三部曲) |
four | quadr/quart | tetra | quadrangle(四边形), quadruped(四足动物), quadruple(四倍), quarter(四分之一), quartet(四重奏); tetragon(四边形), tetrahedron(四面体), tetralogy(四部曲) |
five | quint | penta | quintuple(五倍), quintet(五重奏), quincentenary(五百周年纪念的); pentagon(五角大楼), pentathlon(五项全能), pentad(五个一组；五价元素) |
six | sex(t) | hexa | sexangle, sextet(六重奏), sexfoil(六折的); hexagon, hexahedron(六面体), hexapod(六足虫，昆虫) |
seven | sept | hept(a) | septangle, septennial(七年一度的), septavalent(七价的); heptagon, heptahedron, heptachord(七弦琴) |
eight | octo | octo | octopus(章鱼), octapod(章鱼类生物), octuple(八倍), octagon, octet(八人合唱团), octahedron |
nine | nov/non | ennea | nonagon, nonet(九重奏), nonuple(九倍); ennead(九个一组的), enneagon, enneasyllable |
ten | dec | deca/deka | **(deci-, 十分之一) ** decimalism(十进制), decimeter, decigram, decibel(分贝), decimate(取十分之一；大量毁灭，严重破坏); decade(十年), decagon, decaliter(十升), decathlon(十项全能运动) |
half | semi | hemi | semicircle, semicolonial(半殖民地的，colony殖民地), semiconductor, semi-finals; hemisphere, hemicycle, hemiplegia(偏瘫) |
one hundred | cent | hect(o) | (centi-, 百分之一) century, centigrade(摄氏度), centimeter; hectogram(百克), hectoliter(百升), hectometer(百米) |
one thousand | mill | kilo | (milli-, 千分之一) millennium(千年，千周年纪念日), milligram(毫克), millipede(马陆，千足虫), millennial(一千年的，千禧年); kilowatt, kilometer |
ten hundred thousand | mega | mega | megawatt, megaton(百万吨) |
many | multi | poly | multitude(多数，大量), multimedia, multifunctional(多功能的); polycentric(多元的), polygon(多边形), polyglot(通晓多种语言的) |
English | Chinese |
---|---|
single | 1 |
double | 2 |
triple | 3 |
quadruple | 4 |
quintuple | 5 |
sextuple | 6 |
septuple | 7 |
octuple | 8 |
nonuple | 9 |
decuple | 10 |
deci | 1/10 |
centi | 1/100 |
milli (eg: millimeter) | 1/1000 |
Appendix 2. 参考文献
https://zhuanlan.zhihu.com/p/22890294
官方quantitative复习资料传送：http://www.ets.org/s/gre/pdf/gre_math_review.pdf
https://www.ets.org/gre/khan
英语数字前缀：https://wenku.baidu.com/view/9d309a60a98271fe910ef946.html
盒须图：https://en.wikipedia.org/wiki/Box_plot