📚 IB Mathematics: Core Algebra Topics | IB数学：代数核心知识点梳理
Algebra forms the backbone of the IB Mathematics curriculum, appearing in both Analysis and Approaches (AA) and Applications and Interpretation (AI). Mastering these core topics is essential for solving problems across all other branches of mathematics, from calculus to statistics. This article provides a structured review of the key algebraic concepts every IB student must know.
代数是 IB 数学课程的核心支柱，无论是分析与方法（AA）还是应用与解释（AI），都离不开它。掌握这些核心知识点是解决微积分、统计等其他数学分支问题的基础。本文将对 IB 学生必须掌握的代数核心概念进行系统梳理。
1. Exponent Rules and Logarithms | 指数法则与对数
Exponent rules simplify expressions involving powers. The most important ones include: aᵐ × aⁿ = aᵐ⁺ⁿ, aᵐ ÷ aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ, and a^(1/n) = ⁿ√a.
指数法则用于化简含幂的表达式。最重要的法则包括：aᵐ × aⁿ = aᵐ⁺ⁿ，aᵐ ÷ aⁿ = aᵐ⁻ⁿ，(aᵐ)ⁿ = aᵐⁿ，a⁰ = 1，a⁻ⁿ = 1/aⁿ，以及 a^(1/n) = ⁿ√a。
Logarithms are the inverse of exponentiation. The definition b = logₐx means aᵇ = x. Key properties include logₐ(xy) = logₐx + logₐy, logₐ(x/y) = logₐx − logₐy, and logₐ(xᵏ) = k logₐx.
对数是幂运算的逆运算。定义 b = logₐx 意味着 aᵇ = x。重要性质包括：logₐ(xy) = logₐx + logₐy，logₐ(x/y) = logₐx − logₐy，以及 logₐ(xᵏ) = k logₐx。
The change of base rule is useful for evaluating logarithms in different bases: logₐb = logₓb / logₓa, where x is any valid base (often 10 or e).
换底公式可用于在不同底数之间转换对数：logₐb = logₓb / logₓa，其中 x 可以是任意合法底数（通常取 10 或 e）。
2. Quadratic Equations and Functions | 二次方程与二次函数
A quadratic function is written as f(x) = ax² + bx + c, where a ≠ 0. Its graph is a parabola, and its roots can be found by factoring, completing the square, or using the quadratic formula.
二次函数写成 f(x) = ax² + bx + c，其中 a ≠ 0。其图像为抛物线，可通过因式分解、配方法或求根公式求解根。
x = (−b ± √(b² − 4ac)) / (2a)
The discriminant Δ = b² − 4ac determines the nature of the roots: if Δ > 0, two distinct real roots; if Δ = 0, one repeated real root; if Δ < 0, two complex conjugate roots.
判别式 Δ = b² − 4ac 决定根的性质：若 Δ > 0，有两个不等实根；若 Δ = 0，有一个重根；若 Δ < 0，有两个共轭复根。
For a quadratic function, the axis of symmetry is x = −b/(2a), and the vertex is located at this x-value. The sum and product of the roots are −b/a and c/a respectively.
二次函数的对称轴为 x = −b/(2a)，顶点在该 x 值处。两根之和为 −b/a，两根之积为 c/a。
3. Systems of Equations | 方程组
Linear simultaneous equations in two variables can be solved by substitution, elimination, or graphically. For example, solving 2x + 3y = 8 and x − y = 1 yields a unique solution if the lines intersect.
二元线性方程组可通过代入法、消元法或图像法求解。例如，解 2x + 3y = 8 与 x − y = 1，若两直线相交，则有唯一解。
For three variables or more, matrix methods such as Gaussian elimination or inverse matrices are often used. In IB AA, students also solve systems represented as matrix equations AX = B.
对于三个或更多变量，常使用矩阵方法，如高斯消元法或逆矩阵法。在 IB AA 中，学生还需解以矩阵方程 AX = B 表示的方程组。
A system may have one unique solution, infinitely many solutions, or no solution. When the determinant of the coefficient matrix is zero, the system is either inconsistent or dependent.
方程组可能有一个唯一解、无穷多解或无解。当系数矩阵的行列式为零时，方程组要么无解，要么有无穷多解。
4. Inequalities | 不等式
Solving linear inequalities follows the same rules as equations, except that multiplying or dividing by a negative number reverses the inequality sign.
解线性不等式与解方程规则类似，唯一需要注意的是：乘以或除以负数时，不等号方向要改变。
Quadratic inequalities are solved by finding critical roots and testing intervals. For example, x² − 4 > 0 gives x < −2 or x > 2.
二次不等式通过求临界根并检验区间来解。例如，x² − 4 > 0 的解为 x < −2 或 x > 2。
Rational inequalities require care with undefined points. For instance, solving (x − 1)/(x + 2) > 0 involves examining the sign changes around x = 1 and x = −2.
分式不等式需要特别注意未定义点。例如，解 (x − 1)/(x + 2) > 0 需要考察 x = 1 和 x = −2 附近的符号变化。
5. Sequences and Series | 数列与级数
An arithmetic sequence has a common difference d. The nth term is uₙ = u₁ + (n − 1)d, and the sum of the first n terms is Sₙ = n/2 [2u₁ + (n − 1)d] = n/2 (u₁ + uₙ).
等差数列有公差 d。通项公式为 uₙ = u₁ + (n − 1)d，前 n 项和为 Sₙ = n/2 [2u₁ + (n − 1)d] = n/2 (u₁ + uₙ)。
A geometric sequence has a common ratio r. The nth term is uₙ = u₁rⁿ⁻¹, and the sum is Sₙ = u₁(1 − rⁿ)/(1 − r) for r ≠ 1.
等比数列有公比 r。通项公式为 uₙ = u₁rⁿ⁻¹，前 n 项和为 Sₙ = u₁(1 − rⁿ)/(1 − r)，其中 r ≠ 1。
For an infinite geometric series with |r| < 1, the sum converges to S∞ = u₁/(1 − r). This is essential in financial mathematics and recurring decimals.
当 |r| < 1 时，无穷等比级数收敛于 S∞ = u₁/(1 − r)。这一公式在金融数学和循环小数中至关重要。
6. Binomial Theorem | 二项式定理
The binomial theorem expands (a + b)ⁿ for positive integer n. The expansion is (a + b)ⁿ = C(n,0)aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + … + C(n,n)bⁿ.
二项式定理用于展开 (a + b)ⁿ，其中 n 为正整数。展开式为 (a + b)ⁿ = C(n,0)aⁿ + C(n,1)aⁿ⁻¹b + C(n,2)aⁿ⁻²b² + … + C(n,n)bⁿ。
The binomial coefficients C(n,r) can be calculated as n! / [r!(n − r)!], and they follow Pascal’s triangle. In IB, you must also find a specific term, such as the term independent of x.
二项式系数 C(n,r) 可通过 n! / [r!(n − r)!] 计算，并遵循帕斯卡三角。在 IB 考试中，还需能找到特定项，例如常数项。
C(n,r) = nC r = n! / (r! (n − r)!)
The general term in the expansion is T₍ᵣ₊₁₎ = C(n,r) aⁿ⁻ʳ bʳ. This helps in finding coefficients without expanding the whole expression.
展开式中的通项为 T₍ᵣ₊₁₎ = C(n,r) aⁿ⁻ʳ bʳ。这有助于不求整体展开而直接求某一项的系数。
7. Functions and Transformations | 函数与变换
Algebraic understanding of functions includes domain, range, composition, and inverse functions. A function f maps each input to exactly one output; its inverse f⁻¹ satisfies f(f⁻¹(x)) = x.
函数的代数理解包括定义域、值域、复合函数和反函数。函数 f 将每个输入映射到唯一输出；其反函数 f⁻¹ 满足 f(f⁻¹(x)) = x。
Transformations of graphs follow specific algebraic rules: y = f(x) + a shifts upward, y = f(x − b) shifts right, y = −f(x) reflects in the x-axis, and y = f(−x) reflects in the y-axis.
图像变换遵循特定代数规则：y = f(x) + a 表示向上平移，y = f(x − b) 表示向右平移，y = −f(x) 表示关于 x 轴对称，y = f(−x) 表示关于 y 轴对称。
Vertical stretch: y = k f(x) stretches by factor k; horizontal stretch: y = f(kx) compresses horizontally by factor 1/k. Understanding these helps in sketching complex functions.
纵向伸缩：y = k f(x) 拉伸 k 倍；横向伸缩：y = f(kx) 横向压缩 1/k 倍。理解这些有助于绘制复杂函数图像。
8. Complex Numbers | 复数
Complex numbers are written as z = a + bi, where a, b are real and i² = −1. The conjugate is z̄ = a − bi, and the modulus is |z| = √(a² + b²).
复数写成 z = a + bi，其中 a、b 为实数，i² = −1。共轭复数为 z̄ = a − bi，模为 |z| = √(a² + b²)。
Addition and subtraction are performed component-wise: (a + bi) + (c + di) = (a + c) + (b + d)i. Multiplication is similar to expanding binomials:
加减法按实部虚部分别进行：(a + bi) + (c + di) = (a + c) + (b + d)i。乘法类似于展开二项式：
(a + bi)(c + di) = (ac − bd) + (ad + bc)i
Division uses the conjugate to rationalise the denominator: (a + bi)/(c + di) = (a + bi)(c − di)/(c² + d²). Complex roots of quadratic equations often appear in conjugate pairs.
除法利用共轭复数化简分母：(a + bi)/(c + di) = (a + bi)(c − di)/(c² + d²)。二次方程的复数根通常成共轭对出现。
9. Matrices | 矩阵
A matrix is a rectangular array of numbers. Matrix addition and subtraction require same dimensions; scalar multiplication multiplies every entry by the scalar.
矩阵是按矩形排列的数的阵列。矩阵加减法要求维度相同；标量乘法将每个元素乘以该标量。
Matrix multiplication is defined as row-by-column combination. The product of a 2×2 matrix A and B is:
矩阵乘法定义为行乘以列的组合。2×2 矩阵 A 与 B 的乘积为：
AB = [[a,b],[c,d]] × [[e,f],[g,h]] = [[ae+bg, af+bh],[ce+dg, cf+dh]]
For a 2×2 matrix A = [[a,b],[c,d]], the determinant is det(A) = ad − bc. If det(A) ≠ 0, the inverse is A⁻¹ = (1/(ad − bc)) × [[d,−b],[−c,a]].
对于 2×2 矩阵 A = [[a,b],[c,d]]，行列式为 det(A) = ad − bc。若 det(A) ≠ 0，则逆矩阵为 A⁻¹ = (1/(ad − bc)) × [[d,−b],[−c,a]]。
10. Algebraic Simplification and Polynomials | 代数化简与多项式
Algebraic simplification involves expanding, factorising, and simplifying rational expressions. Common identities include (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².
代数化简涉及展开、因式分解和简化分式。常见恒等式包括 (a + b)² = a² + 2ab + b² 和 (a − b)² = a² − 2ab + b²。
The factor theorem states that (x − a) is a factor of polynomial P(x) if and only if P(a) = 0. Synthetic division can be used to factorise higher-degree polynomials.
因式定理指出：当且仅当 P(a) = 0 时，(x − a) 是多项式 P(x) 的因式。综合除法可用于分解高次多项式。
Solving polynomial equations often requires combining the factor theorem, long division, and quadratic techniques. The fundamental theorem of algebra guarantees that a degree-n polynomial has exactly n roots over the complex numbers.
解多项式方程通常需要结合因式定理、长除法和二次技巧。代数基本定理保证 n 次多项式在复数范围内恰有 n 个根。
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