📚 American High School Math Core Vocabulary | 美高数学核心词汇汇总
Mastering mathematical terminology is essential for success in American high school math courses, from Algebra I to AP Calculus. This bilingual vocabulary guide provides clear English definitions alongside accurate Chinese translations, helping students bridge language gaps and strengthen conceptual understanding. Use it as a quick reference to boost your confidence in class, on homework, and during exams.
掌握数学术语是学好美高数学课程（从代数I到AP微积分）的关键。这份双语词汇指南提供了清晰的英文定义和准确的中文翻译，帮助学生跨越语言障碍，加深对概念的理解。你可以把它当作速查手册，提升课堂、作业和考试中的自信。
1. Basic Arithmetic and Number Systems | 基础算术与数系
Integer: a whole number that can be positive, negative, or zero, without any fractional or decimal part.
整数：可以为正、负或零的完整数字，不包含分数或小数部分。
Rational number: any number that can be expressed as a fraction a/b where a and b are integers and b is not zero.
有理数：任何可以表示为分数 a/b 的数，其中 a 和 b 为整数且 b 不为零。
Irrational number: a number that cannot be written as a simple fraction; its decimal form is non-repeating and non-terminating, e.g., sqrt2, pi.
无理数：不能表示为简单分数的数；其小数形式无限不循环，例如 sqrt2、pi。
Real number: the set of all rational and irrational numbers, represented on the number line.
实数：所有有理数和无理数的集合，可以在数轴上表示。
Prime number: a natural number greater than 1 that has no positive divisors other than 1 and itself.
质数（素数）：大于1的自然数，且只有1和自身两个正因数。
Absolute value: the distance of a number from zero on the number line, always non-negative.
绝对值：一个数在数轴上到零的距离，永远非负。
2. Algebraic Expressions and Equations | 代数表达式与方程
Variable: a symbol, usually a letter, that represents an unknown or changing value.
变量：通常用字母表示的符号，代表未知或可变化的量。
Coefficient: a numerical factor multiplying a variable in an algebraic term, e.g., 3 in 3x².
系数：代数项中与变量相乘的数字因子，例如 3x² 中的 3。
Expression: a combination of numbers, variables, and operation symbols without an equality sign.
表达式：由数字、变量和运算符号组成的式子，不含等号。
Equation: a mathematical statement that two expressions are equal, indicated by '='.
方程：表明两个表达式相等的数学陈述，用“=”表示。
Like terms: terms that have the same variable raised to the same power; they can be combined by adding coefficients.
同类项：变量及其指数都相同的项；可以通过系数相加来合并。
Polynomial: an expression consisting of variables and coefficients involving only addition, subtraction, multiplication, and non-negative integer exponents.
多项式：由变量和系数通过加法、减法、乘法及非负整数指数组成的表达式。
3. Functions and Graphs | 函数与图像
Function: a relation where each input (x) has exactly one output (y).
函数：每个输入值 (x) 恰好对应一个输出值 (y) 的关系。
Domain: the set of all possible input values (x-values) for which the function is defined.
定义域：函数有定义的所有可能输入值（x值）的集合。
Range: the set of all possible output values (y-values) that the function can produce.
值域：函数可能产生的所有输出值（y值）的集合。
Intercept: a point where a graph crosses the x-axis (x-intercept) or y-axis (y-intercept).
截距：图像与x轴的交点（x截距）或与y轴的交点（y截距）。
Slope: a measure of the steepness of a line, typically calculated as rise over run, delta y / delta x.
斜率：直线陡峭程度的度量，通常计算为纵增量除以横增量，Δy / Δx。
Asymptote: a line that a graph approaches but never touches; can be horizontal, vertical, or slanted.
渐近线：图像无限逼近但永不相交的直线；可分为水平、垂直或斜渐近线。
4. Geometry: Shapes and Properties | 几何：形状与性质
Congruent: having the same shape and size; corresponding sides and angles are equal.
全等：形状和大小完全相同；对应边和对应角都相等。
Similar: having the same shape but not necessarily the same size; corresponding angles are equal and sides are proportional.
相似：形状相同但大小不一定相同；对应角相等，对应边成比例。
Hypotenuse: the longest side of a right triangle, opposite the right angle.
斜边：直角三角形中最长的边，即直角的对边。
Perimeter: the total distance around the boundary of a two-dimensional shape.
周长：围绕二维图形边界一周的总距离。
Area: the amount of space enclosed within a two-dimensional shape.
面积：二维图形内部所包含的空间大小。
Volume: the amount of space occupied by a three-dimensional object.
体积：三维物体所占空间的大小。
5. Trigonometry | 三角学
Sine (sin): in a right triangle, the ratio of the length of the opposite side to the hypotenuse.
正弦 (sin)：在直角三角形中，对边长度与斜边长度的比值。
Cosine (cos): in a right triangle, the ratio of the adjacent side to the hypotenuse.
余弦 (cos)：在直角三角形中，邻边长度与斜边长度的比值。
Tangent (tan): in a right triangle, the ratio of the opposite side to the adjacent side; also equal to sin/cos.
正切 (tan)：在直角三角形中，对边长度与邻边长度的比值；也等于 sin/cos。
Radian: a unit of angular measure where one radian is the angle subtended by an arc equal in length to the radius.
弧度：角度度量单位，1 弧度是弧长等于半径时所对的圆心角。
Unit circle: a circle with radius 1 centered at the origin, used to define trigonometric functions for all angles.
单位圆：以原点为圆心、半径为1的圆，用于定义任意角的三角函数。
Law of Sines / Law of Cosines: formulas relating the sides and angles of any triangle, used for solving non-right triangles.
正弦定理 / 余弦定理：联系任意三角形各边和角的公式，用于解非直角三角形。
6. Statistics and Probability | 统计与概率
Mean: the average of a data set, found by summing all values and dividing by the number of values.
平均值（均数）：数据集的平均数，计算方法为所有数值之和除以数值的个数。
Median: the middle value when a data set is ordered from least to greatest.
中位数：将数据集从小到大排序后位于中间的值。
Mode: the value that appears most frequently in a data set.
众数：数据集中出现次数最多的值。
Standard deviation: a measure of how spread out the data is from the mean.
标准差：衡量数据相对于平均值的离散程度的指标。
Probability: a number between 0 and 1 that indicates the likelihood of an event occurring.
概率：介于0和1之间的数，表示事件发生的可能性大小。
Independent events: events where the outcome of one does not affect the outcome of the other.
独立事件：一个事件的结果不影响另一个事件结果的事件。
7. Sequences and Series | 数列与级数
Arithmetic sequence: a sequence where the difference between consecutive terms is constant.
等差数列：相邻两项的差为常数的数列。
Geometric sequence: a sequence where each term is found by multiplying the previous term by a constant ratio.
等比数列：每一项都是前一项乘以一个常数比率得到的数列。
Series: the sum of the terms of a sequence.
级数：数列各项之和。
Convergence: the property of an infinite series approaching a finite limit as more terms are added.
收敛：无穷级数随着项数增多而趋近于一个有限极限的性质。
Sigma notation: a way to write a sum using the capital Greek letter Sigma to indicate the index and limits.
求和符号：用希腊大写字母 Sigma 表示求和的写法，标明了指数和上下限。
8. Exponents, Radicals, and Logarithms | 指数、根式与对数
Exponent ipt number that shows how many times the base is multiplied by itself, e.g., 2⁴ = 2×2×2×2.
指数：上标数字，表示底数自乘的次数，例如 2⁴ = 2×2×2×2。
Radical: the symbol sqrt used to denote roots, such as square root or cube root.
根式：表示方根的符号 sqrt，如平方根或立方根。
Logarithm: the inverse of exponentiation; logₐ(b) = c means aᶜ = b.
对数：指数运算的逆运算；logₐ(b) = c 意味着 aᶜ = b。
Natural log (ln): a logarithm with base e (approx. 2.71828).
自然对数 (ln)：以 e（约 2.71828）为底的对数。
Change of base formula: logₐ(b) = log(b) / log(a) using common or natural logs.
换底公式：logₐ(b) = log(b) / log(a)，可以使用常用对数或自然对数。
9. Coordinate Geometry and Vectors | 坐标几何与向量
Cartesian coordinate system: a grid defined by a horizontal x-axis and a vertical y-axis intersecting at the origin (0,0).
笛卡尔坐标系：由水平的x轴和垂直的y轴在原点 (0,0) 相交构成的网格。
Distance formula: the distance between points (x₁,y₁) and (x₂,y₂) is sqrt[(x₂−x₁)² + (y₂−y₁)²].
距离公式：点 (x₁,y₁) 和 (x₂,y₂) 之间的距离为 sqrt[(x₂−x₁)² + (y₂−y₁)²]。
Midpoint: the point halfway between two endpoints, calculated as ((x₁+x₂)/2, (y₁+y₂)/2).
中点：两点正中间的点，计算公式为 ((x₁+x₂)/2, (y₁+y₂)/2)。
Vector: a quantity that has both magnitude and direction, often represented by an arrow.
向量：既有大小又有方向的量，常用箭头表示。
Dot product: an operation that takes two vectors and returns a scalar; related to the cosine of the angle between them.
点积（内积）：取两个向量并返回标量的运算；与两向量夹角的余弦相关。
10. Calculus Basics | 微积分基础
Limit: the value that a function or sequence approaches as the input approaches some value.
极限：当输入趋近某个值时，函数或数列趋近的值。
Derivative: the instantaneous rate of change of a function, or the slope of the tangent line at a point.
导数：函数的瞬时变化率，或函数在某点切线的斜率。
Integral: the accumulation of quantities; can represent area under a curve or the antiderivative.
积分：量的累积；可表示曲线下的面积或反导数。
Continuity: a function is continuous if there are no breaks, jumps, or holes in its graph.
连续性：如果函数图像没有间断、跳跃或空洞，则该函数连续。
Chain rule: a rule for differentiating composite functions: (f(g(x)))' = f'(g(x)) * g'(x).
链式法则：用于复合函数求导的规则：(f(g(x)))' = f'(g(x)) * g'(x)。
Fundamental Theorem of Calculus: connects differentiation and integration, stating that they are inverse processes.
微积分基本定理：将微分与积分联系起来，说明它们是互逆的运算。
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