📚 Essential Mathematical Formulas (Basic) | 数学常用公式汇总（基础篇）
Mastering basic mathematical formulas is the first step toward success in any mathematics course. Formulas are the vocabulary of the subject: once they are automatic, you can spend your mental energy on reasoning instead of recall. This article compiles the essential formulas from algebra, indices, logarithms, sequences, coordinate geometry, trigonometry, binomial expansion and introductory calculus, and explains where each one is used. Whether you are preparing for an examination or refreshing long-forgotten knowledge, keep this page as a quick reference.
掌握基础数学公式是学好数学的第一步。公式就像这门学科的词汇，一旦熟练到不需思考，你就能把精力放在推理而不是回忆上。本文汇总了代数、指数、对数、数列、坐标几何、三角、二项式展开与入门微积分中的常用公式，并说明每一个公式的使用场景。无论你是在备考还是巩固旧知识，都可以把这一页当作速查手册。
1. Algebraic Identities | 代数恒等式
Algebraic identities are equations that hold true for all values of the variables. They let you expand brackets, factorise expressions and simplify fractions quickly, and they appear in almost every algebra question.
代数恒等式是对所有变量取值都成立的等式，可以用于快速展开括号、因式分解和化简分式，几乎每道代数题都会用到。
(a + b)² = a² + 2ab + b²
Square of a sum: the middle term 2ab is the one students most often forget.
和的平方：中间项 2ab 是最容易被漏掉的一项。
(a − b)² = a² − 2ab + b²
Square of a difference: only the sign of the middle term changes.
差的平方：只有中间项的符号发生变化。
a² − b² = (a − b)(a + b)
Difference of squares: the standard tool for rationalising and for cancelling factors.
平方差：分母有理化和约分的标准工具。
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)
Sum and difference of cubes: note that the quadratic factor is never further factorisable over the reals.
立方和与立方差：注意后面那个二次因式在实数范围内无法再分解。
(a + b)³ = a³ + 3a²b + 3ab² + b³
(a − b)³ = a³ − 3a²b + 3ab² − b³
Binomial cubes: the coefficients 1, 3, 3, 1 come straight from Pascal's triangle.
立方展开：系数 1、3、3、1 直接来自杨辉三角。
(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca
Three-term square: useful when completing the square in two variables or expanding circle equations.
三项平方：在处理双变量配方或展开圆的方程时很有用。
2. Quadratic Formula and Discriminant | 二次方程求根公式与判别式
h a ≠ 0, the solutions are given by the quadratic formula, and the discriminant tells you what kind of roots to expect before you calculate anything.
对于标准形式 a x² + b x + c = 0（a ≠ 0）的二次方程，其解可由求根公式给出；而判别式能让你在动笔计算之前就预判根的性质。
x = (−b ± √b² − 4ac) / (2a)
Δ = b² − 4ac
ed real root; if Δ < 0 there are no real roots and the curve does not cross the x-axis.
若 Δ > 0，有两个不相等的实根；若 Δ = 0，有一个重实根；若 Δ < 0，没有实根，抛物线与 x 轴不相交。
x₁ + x₂ = −b / a   and   x₁ · x₂ = c / a
Vieta's formulas give the sum and product of the roots without solving the equation.
韦达定理给出了两根之和与两根之积，无需解方程即可使用。
y = a(x − h)² + k,  h = −b / (2a),  k = c − b² / (4a)
Completing the square rewrites the quadratic in vertex form, so the turning point (h, k) is read off directly.
配方法把二次式改写为顶点式，可以直接读出顶点坐标 (h, k)。
3. Laws of Indices and Surds | 指数定律与根式
Indices follow a small set of rules that make powers easy to manipulate. The same rules extend to fractional and negative powers, which is exactly how surds are written in index form.
指数遵循一组简明的运算规律，使幂的运算变得容易。这些规律同样适用于分数指数和负指数，而根式其实就是分数指数的另一种写法。
aᵐ × aⁿ = aᵐ⁺ⁿ  aᵐ ÷ aⁿ = aᵐ⁻ⁿ
Multiplying adds the indices; dividing subtracts them, provided a ≠ 0.
同底数相乘指数相加，相除指数相减，前提是 a ≠ 0。
(aᵐ)ⁿ = aᵐⁿ  (ab)ⁿ = aⁿ bⁿ  (a / b)ⁿ = aⁿ / bⁿ
Power of a power multiplies the indices; powers distribute over products and quotients.
幂的幂把指数相乘；积与商的乘方可以分别对每个因式取幂。
a⁰ = 1  (a ≠ 0)   a⁻ⁿ = 1 / aⁿ a1/n = ⁿ√a
Any non-zero base to the power zero equals one; negative indices mean reciprocals; a fractional index means a root.
任何非零数的零次幂都等于 1；负指数表示倒数；分数指数表示开方。
√a × √b = √ab √a ÷ √b = √a / b
Roots can be combined under one radical sign, which is the key to simplifying surds.
根号下的数可以合并到一个根号内，这是化简根式的关键。
1 / √a = √a / a 1 / (a + √b) = (a − √b) / (a² − b)
Rationalising the denominator uses the difference of squares identity to remove surds from the bottom of a fraction.
分母有理化利用平方差公式，把分母中的根号消掉。
4. Laws of Logarithms | 对数运算律
A logarithm answers the question 'to what power must the base be raised?' Every index law has a matching logarithm law, which is why the two topics are always taught together.
对数是回答"底数要乘方多少次"这个问题。每一条指数定律都有对应的对数运算律，所以这两个主题总是放在一起讲。
if aˣ = y then logₐ y = x  (a > 0, a ≠ 1, y > 0)
The definition converts between exponential and logarithmic form; the domain conditions must always be stated.
对数定义实现了指数式与对数式的互换，使用前必须写清定义域条件。
logₐ (mn) = logₐ m + logₐ n
logₐ (m / n) = logₐ m − logₐ n
logₐ (mᵏ) = k · logₐ m
The three workhorse laws: a product becomes a sum, a quotient becomes a difference, and a power becomes a multiplier.
三条核心运算律：乘积变加法，商变减法，幂次变系数。
logₐ a = 1   logₐ 1 = 0  alogₐ x = x
These three results are frequently needed when solving equations involving logs.
解对数方程时经常会用到这三条结论。
logₐ b = logₙ b / logₙ a
The change-of-base rule lets you evaluate any logarithm on a calculator that only has log and ln buttons.
换底公式让你只用计算器上的 log 和 ln 键就能求出任意底数的对数值。
5. Linear Equations, Simultaneous Equations and Inequalities | 一次方程、联立方程与不等式
Linear relationships underpin most applied questions in mechanics, business and coordinate geometry. Solving them reliably and handling inequality signs correctly are basic exam skills.
一次关系是力学、经济与坐标几何应用题的基础。可靠地解方程并且正确处理不等号，是考试中的基本技能。
a x + b = 0  →  x = −b / a  (a ≠ 0)
A single linear equation always has exactly one solution when a ≠ 0.
当 a ≠ 0 时，一元一次方程总有且仅有一个解。
y = m x + c
Gradient-intercept form: m is the slope and c is the y-intercept.
斜截式：m 是斜率，c 是 y 轴截距。
|x| < a  →  −a < x < a |x| > a  →  x < −a or x > a
Modulus inequalities split into a single interval or two separate rays, and the direction of the sign decides which.
绝对值不等式要么化为一个区间，要么化为两条射线，由不等号方向决定。
For quadratic inequalities, factorise first, find the critical values, then test a value in each interval to decide where the expression is positive or negative.
解二次不等式时，先因式分解求出临界值，再在每个区间内取测试值，判断表达式的正负。
multiplying or dividing by a negative number reverses the inequality sign
This single rule causes more lost marks than any other in the topic, so check the sign every time.
这一条规则造成的失分比其他任何错误都多，所以每次变形都要检查符号。
6. Sequences and Series | 数列与级数
Arithmetic and geometric sequences are the two standard patterns tested at introductory level. Knowing both the nth-term and the sum formulas saves a great deal of time.
等差数列与等比数列是基础阶段考查的两种标准模式。同时掌握通项公式与求和公式能节省大量时间。
Arithmetic:  aₙ = a₁ + (n − 1)d
Sₙ = n/2 · [2a₁ + (n − 1)d] = n/2 · (a₁ + aₙ)
The common difference d is found by subtracting any term from the next one.
公差 d 等于相邻两项中后项减前项。
Geometric:  aₙ = a₁ rⁿ⁻¹
Sₙ = a₁ (1 − rⁿ) / (1 − r)  (r ≠ 1)
The common ratio r is found by dividing any term by the previous one.
公比 r 等于任意一项除以前一项。
S∞ = a₁ / (1 − r)  only when |r| < 1
A geometric series converges only when the ratio is strictly between −1 and 1; otherwise
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图