AS CIE Further Mathematics: Quick Memorisation of Key Terms | AS CIE 进阶数学：词汇术语速记指南

📚 AS CIE Further Mathematics: Quick Memorisation of Key Terms | AS CIE 进阶数学：词汇术语速记指南

AS CIE Further Mathematics brings a host of new concepts and a specialised vocabulary that can feel daunting. This rapid memorisation guide unpacks the essential terms from the AS Further Pure 1 syllabus, pairing clear English definitions with Chinese explanations and memory hooks. Strengthening your terminology base is the first step towards confident problem‑solving.

AS CIE 进阶数学引入了大量全新概念和专业术语，容易让人感到无从下手。本速记指南拆解了 AS 进阶纯数 1 的核心词汇，将清晰的英文定义与中文解释、记忆窍门一一对应。打好术语基础，是自信解题的第一步。

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1. Complex Numbers | 复数

A complex number is written z = a + bi, where a, b are real numbers and the imaginary unit i satisfies i² = -1. We denote the real part by Re(z) = a and the imaginary part by Im(z) = b. Note that Im(z) is the real coefficient b, not bi.

复数写作 z = a + bi，其中 a, b 是实数，虚数单位 i 满足 i² = -1。实部记为 Re(z) = a，虚部记为 Im(z) = b。注意 Im(z) 是实系数 b，而不是 bi。

The conjugate of z is z* = a - bi (also written z̄ al number. It is useful for division: multiply numerator and denominator by the conjugate.

共轭复数 定义为 z* = a - bi（也写作 z̄）。共轭将复数关于实轴反射，并且满足 z z* = a² + b² 为实数。复数除法中常用它：分子分母同乘分母的共轭。

The modulus of z is |z| = √a² + b², giving the distance from the origin in the complex plane. The argument, arg z = θ, satisfies tan θ = b/a taking the correct quadrant. The principal argument lies in (-π, π].

模（绝对值）为 |z| = √a² + b²，表示复平面上点到原点的距离。辐角 arg z = θ 满足 tan θ = b/a 并需根据象限确定。主辐角通常取 (-π, π]。

Polar form: z = r (cos θ + i sin θ) with r = |z|. Multiplication then becomes "multiply moduli, add arguments". De Moivre's theorem cos θ + i sin θ))ⁿ = rⁿ (cos nθ + i sin nθ). This is a powerful tool for powers and roots.

极坐标形式：z = r (cos θ + i sin θ)，其中 r = |z|。此时复数乘法规则为『模相乘，辐角相加』。德莫弗定理： (r (cos θ + i sin θ))ⁿ = rⁿ (cos nθ + i sin nθ)，是处理乘方和开方的利器。

The exponential form is z = r e^iθ, thanks to Euler's formula e^iθ = cos θ + i sin θ. This compactly links complex numbers, trigonometry and exponentials.

指数形式 z = r e^iθ 源自欧拉公式 e^iθ = cos θ + i sin θ，它精妙地联结了复数、三角与指数。

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2. Matrices | 矩阵

A matrix is a rectangular array of numbers. Its order or dimension is rows × columns, e.g. a 2×2 matrix. We write A for a matrix and often use bold or capital letters.

矩阵是一个矩形的数字阵列。它的阶数或维度表示为 行数×列数，例如 2×2 矩阵。常用大写字母 A 表示矩阵。

The determinant of a 2×2 matrix A = [[a, b], [c, d]] is det A = ad - bc. A matrix with zero determinant is called singular and has no inverse; a matrix with non‑zero determinant is non‑singular.

2×2 矩阵 A = [[a, b], [c, d]] 的行列式为 det A = ad - bc。行列式为零的矩阵称为奇异矩阵，不可逆；行列式非零的矩阵称非奇异矩阵。

The inverse of a 2×2 non‑singular matrix is A⁻¹ = 1/(ad - bc) [[d, -b], [-c, a]]. Memory trick: swap the main diagonal entries, change signs of the off‑diagonal entries, and divide by the determinant.

2×2 非奇异矩阵的逆矩阵为 A⁻¹ = 1/(ad - bc) [[d, -b], [-c, a]]。记忆口诀：主对角线元素交换位置，副对角线元素变号，再除以行列式。

The transpose of A, denoted Aᵀ, swaps rows and columns. Matrix multiplication is not commutative: generally AB ≠ BA. The identity matrix I (e.g. I₂ = [[1,0],[0,1]]) satisfies AI = IA = A.

转置矩阵 Aᵀ 将行与列互换。矩阵乘法不满足交换律：通常 AB ≠ BA。单位矩阵 I（如 I₂ = [[1,0],[0,1]]）满足 AI = IA = A。

Matrices represent linear transformations ors in the plane. The determinant tells you the area scale factor of the transformation; a negative determinant indicates a reflection.

矩阵表示线性变换：一个 2×2 矩阵将平面上的列向量进行变换。行列式给出变换对面积缩放的倍数；行列式为负意味着包含反射。

Solving simultaneous equations AX = B can be done by premultiplying by A⁻¹, giving X = A⁻¹B, provided A is non‑singular. Otherwise the system may have no unique solution.

解线性方程组 AX = B 可通过左乘 A⁻¹ 得到 X = A⁻¹B，要求 A 非奇异。若 A 奇异，方程组可能无唯一解。

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3. Polynomials and Their Roots | 多项式及其根

The degree of a polynomial is the highest power of the variable. The Fundamental Theorem of Algebra states that a polynomial of degree n has exactly n complex roots (counting multiplicities).

多项式的次数指变量的最高次幂。代数基本定理指出，n 次多项式恰好有 n 个复数根（计重数）。

For a quadratic ax² + bx + c = 0 with roots α, β, the sum is α + β = -b/a and the product is αβ = c/a. For a cubic ax³ + bx² + cx + d = 0 with roots α, β, γ, we have α+β+γ = -b/a, αβ+βγ+γα = c/a, αβγ = -d/a. These are Vieta's formulas.

对于二次方程 ax² + bx + c = 0，根 α, β 满足 α + β = -b/a，αβ = c/a。三次方程 ax³ + bx² + cx + d = 0 的根 α, β, γ 满足 α+β+γ = -b/a，αβ+βγ+γα = c/a，αβγ = -d/a。这就是韦达定理。

The Factor Theorem: (x - p) is a factor of f(x) if and only if f(p) = 0. The Remainder Theorem: when f(x) is divided by (x - p), the remainder is f(p). These are vital for factorising polynomials.

因子定理：(x - p) 是 f(x) 的因式当且仅当 f(p) = 0。余式定理：f(x) 除以 (x - p) 的余式为 f(p)。这两个定理是多项式因式分解的基石。

The Rational Root Theorem helps identify possible rational roots: for a polynomial with integer coefficients, any rational root p/q in lowest terms has p dividing the constant term and q dividing the leading coefficient.

有理根定理可帮助锁定可能的有理根：对于整系数多项式，任何既约有理根 p/q 满足 p 整除常数项，q 整除首项系数。

If roots are complex, they occur in conjugate pairs when coefficients are real. This symmetry is extremely useful when constructing polynomials from given roots.

若系数为实数，复根总是成对共轭出现。这种对称性在根据已知根构造多项式时极为有用。

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4. Polar Coordinates | 极坐标

In polar coordinates, a point is defined by its distance from the origin r and the angle θ measured from the positive x‑axis. Conversions: x = r cos θ, y = r sin θ, and r = √x² + y², θ = arctan(y/x) (adjusted for quadrant).

极坐标系中，点由到原点的距离 r 和从正 x 轴起算的角度 θ 确定。转换关系：x = r cos θ，y = r sin θ，以及 r = √x² + y²，θ = arctan(y/x)（需根据象限调整）。

A polar curve is given as r = f(θ). Common shapes: r = a (circle), r = a cos θ (circle through the pole), r = a sin θ, and cardioids like r = a(1 + cos θ).

极坐标曲线表示为 r = f(θ)。常见形状：r = a（圆），r = a cos θ（过极点的圆），r = a sin θ，以及心形线如 r = a(1 + cos θ)。

The area bounded by a polar curve and the half‑lines θ = α, θ = β is given by Area = (1/2) ∫(α to β) r² dθ. The square of r is integrated, not r itself.

由极坐标曲线与射线 θ = α、θ = β 围成的面积为 面积 = (1/2) ∫(α→β) r² dθ。积分的是 r²，而不是 r。

mmetry about the pole). Plotting key points for θ = 0, π/2, π, etc. helps build the shape.

画图时可利用对称性检验：若 r(θ) = r(-θ)，图形关于极轴（初始线）对称；若 r(θ) = -r(θ+π)，则关于极点对称。取 θ = 0, π/2, π 等关键点描点可辅助成形。

f(θ) = 0 gives the angles at which the curve passes through the pole, guiding the sketch.

当某个 θ 使 r = 0 时，曲线在该角度通过极点，且切线方向由该 θ 决定。解 f(θ) = 0 即可找到极点的角度，这对草图至关重要。

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5. Hyperbolic Functions | 双曲函数

The hyperbolic functions are defined using exponentials: sinh x = (eˣ - e⁻ˣ)/2, cosh x = (eˣ + e⁻ˣ)/2, and tanh x = sinh x / cosh x. They echo circular trigonometric functions but with important sign differences.

双曲函数由指数函数定义：sinh x = (eˣ - e⁻ˣ)/2，cosh x = (eˣ + e⁻ˣ)/2，tanh x = sinh x / cosh x。它们与圆三角函数形式相似，但有重要符号差异。

Fundamental identity: cosh² x - sinh² x = 1. Compare with cos² θ + sin² θ = 1. Other identities include sinh(2x) = 2 sinh x cosh x and cosh(2x) = cosh² x + sinh² x.

基本恒等式：cosh² x - sinh² x = 1。与 cos² θ + sin² θ = 1 比较可见符号不同。其他恒等式如 sinh(2x) = 2 sinh x cosh x 和 cosh(2x) = cosh² x + sinh² x。

Differentiation: d/dx sinh x = cosh x, d/dx cosh x = sinh x (no minus sign!), d/dx tanh x = sech² x. This makes hyperbolic functions very easy to differentiate repeatedly.

求导：d/dx sinh x = cosh x，d/dx cosh x = sinh x（注意没有负号！），d/dx tanh x = sech² x。这使得双曲函数的反复求导非常简洁。

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