Complex numbers · 复数
| English | 中文 | Pinyin · 拼音 |
|---|---|---|
| complex number/ˈkɒmpleks ˈnʌmbə/ | 复数 | fù shù |
| real part/rɪəl pɑːt/ | 实部 | shí bù |
| imaginary part/ɪˈmædʒɪnəri pɑːt/ | 虚部 | xū bù |
| conjugate/ˈkɒndʒuːɡeɪt/ | 共轭 | gòng è |
| modulus/ˈmɒdjʊləs/ | 绝对值 | jué duì zhí |
| argument/ˈɑːɡjuːmənt/ | 辐角 | fú jiǎo |
| Argand diagram/ˈɑːɡænd ˈdaɪəɡræm/ | 阿干图 | ā gàn tú |
| polar form/ˈpəʊlə fɔːm/ | 极坐标形式 | jí zuò biāo xíng shì |
| discriminant/dɪˈskrɪmɪnənt/ | 判别式 | pàn bié shì |
The number that doesn't exist
- What is $\sqrt{-1}$? For centuries, mathematicians said it doesn't exist. Then they discovered that pretending it does — calling it $i$ — unlocks solutions to equations that otherwise have none.
- Complex numbers 复数 extend the number line into a plane, and they're essential for electrical engineering, quantum mechanics, and signal processing.
不存在的数
- $\sqrt{-1}$ 是什么?几个世纪以来,数学家说它不存在。然后他们发现假装它存在——把它叫做 $i$——能解锁原本无解的方程的解。
- 复数(complex numbers)把数轴扩展成一个平面,而且它们对电气工程、量子力学和信号处理必不可少。
Complex numbers
- A complex number is $z = x + iy$, where $i^2 = -1$ — $x$ is the real part 实部, $y$ the imaginary part 虚部.
- The conjugate 共轭 $z^* = x - iy$. Real-coefficient polynomials have non-real roots in conjugate pairs.
- The modulus 绝对值 $|z| = \sqrt{x^2 + y^2}$; the argument 辐角 is the angle from the positive real axis.
Worked example. $z = 3 + 4i$. Modulus $= \sqrt{9 + 16} = 5$. Conjugate $= 3 - 4i$.
$i^2 = -1$, not $1$. When multiplying complex numbers, remember that $i^2 = -1$. For example, $(2+i)(3+i) = 6 + 2i + 3i + i^2 = 6 + 5i - 1 = 5 + 5i$.
Self-similar patterns like Romanesco arise from iterating in the complex plane
复数
- 一个复数是 $z = x + iy$,其中 $i^2 = -1$——$x$ 是实部(real part),$y$ 是虚部(imaginary part)。
- 共轭(conjugate)$z^* = x - iy$。实系数多项式的非实根成共轭对出现。
- 模(modulus)$|z| = \sqrt{x^2 + y^2}$;辐角(argument)是从正实轴起的角。
算例。 $z = 3 + 4i$。模 $= \sqrt{9 + 16} = 5$。共轭 $= 3 - 4i$。
$i^2 = -1$,不是 $1$。 当乘复数时,记住 $i^2 = -1$。例如,$(2+i)(3+i) = 6 + 2i + 3i + i^2 = 6 + 5i - 1 = 5 + 5i$。
像罗马花椰菜这样的自相似图案来自在复平面中迭代
The Argand diagram · 阿尔冈图
z = x + yi
Plot a complex number as a point: its distance from O is the modulus, its angle is the argument. · 把一个复数画成一个点:它到 O 的距离是模,它的角是辐角。
What is the modulus of the complex number 3 + 4i? · 复数 3 + 4i 的模是多少?
|z| = √(3² + 4²) = √(9 + 16) = √25 = 5. · |z| = √(3² + 4²) = √(9 + 16) = √25 = 5。
The conjugate of z = 5 + 2i is: · z = 5 + 2i 的共轭是:
The conjugate flips the sign of the imaginary part: z* = 5 − 2i. · 共轭翻转虚部的符号:z* = 5 − 2i。
What is the value of i²? · i² 的值是多少?
By definition of the imaginary unit, i² = −1. · 由虚数单位的定义,i² = −1。
(2 + i)(3 + i) = a + bi. What is a (the real part)? · (2 + i)(3 + i) = a + bi。a(实部)是多少?
(2+i)(3+i) = 6 + 2i + 3i + i² = 6 + 5i − 1 = 5 + 5i. Real part = 5. · (2+i)(3+i) = 6 + 2i + 3i + i² = 6 + 5i − 1 = 5 + 5i。实部 = 5。
Argand diagram 阿干图 and polar form 极坐标形式
- Plot $z$ as a point on an Argand diagram (real axis horizontal, imaginary axis vertical).
On an Argand diagram, the modulus |z| is the distance from the origin and the argument is the angle θ.
- Polar form: $z = r(\cos\theta + i\sin\theta) = re^{i\theta}$ — multiplying multiplies moduli and adds arguments.
阿尔冈图与极坐标形式
- 把 $z$ 画成阿尔冈图(Argand diagram)上的一个点(实轴水平,虚轴垂直)。
在一个阿尔冈图上,模 |z| 是到原点的距离,辐角是角 θ。
- 极坐标形式(polar form):$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$——相乘时模相乘、辐角相加。
Division using the conjugate
- To divide, multiply top and bottom by the conjugate of the bottom.
- Example: $\dfrac{3+i}{1-i} = \dfrac{(3+i)(1+i)}{(1-i)(1+i)} = \dfrac{3+3i+i+i^2}{1+1} = \dfrac{2+4i}{2} = 1 + 2i$.
用共轭做除法
- 要除,把分子分母同乘分母的共轭。
- 示例：$\dfrac{3+i}{1-i} = \dfrac{(3+i)(1+i)}{(1-i)(1+i)} = \dfrac{3+3i+i+i^2}{1+1} = \dfrac{2+4i}{2} = 1 + 2i$。
(3+i)/(1−i) = a + bi. What is b (the imaginary part)? · (3+i)/(1−i) = a + bi。b(虚部)是多少?
Multiply by (1+i)/(1+i): (3+i)(1+i)/2 = (3+3i+i−1)/2 = (2+4i)/2 = 1+2i. Imaginary part = 2. · 乘以 (1+i)/(1+i):(3+i)(1+i)/2 = (3+3i+i−1)/2 = (2+4i)/2 = 1+2i。虚部 = 2。
Solving equations with complex roots
- A quadratic with negative discriminant 判别式 has complex roots: $x^2 + 4 = 0 \Rightarrow x = \pm 2i$.
- Real-coefficient polynomials always have non-real roots in conjugate pairs: if $a + bi$ is a root, so is $a - bi$.
allow_no_figure: Complex number arithmetic and the Argand diagram are algebraic/geometric concepts that are effectively conveyed through notation and worked examples.
- Besides Cartesian and polar forms, find square roots of a complex number and sketch loci (e.g. $|z-a|=r$).
求解有复根的方程
- 一个判别式为负的二次式有复根:$x^2 + 4 = 0 \Rightarrow x = \pm 2i$。
- 实系数多项式的非实根总是成共轭对:如果 $a + bi$ 是一个根,那么 $a - bi$ 也是。
allow_no_figure: 复数运算和阿尔冈图是代数/几何概念，通过符号和例题可以有效传达。
- 除直角坐标(Cartesian form)和极坐标外,求复数的平方根(square roots)并画轨迹(loci,如 $|z-a|=r$)。
If a polynomial with real coefficients has root 2 + 3i, it must also have root 2 − 3i. · 如果一个实系数多项式有根 2 + 3i,它也必有根 2 − 3i。
Real-coefficient polynomials always have non-real roots in conjugate pairs. · 实系数多项式的非实根总是成共轭对出现。
You've got it
- $z = x + iy$, $i^2 = -1$; conjugate $z^* = x - iy$; modulus $|z| = \sqrt{x^2+y^2}$
- plot on the Argand diagram; polar form $z = re^{i\theta}$
- divide by multiplying top and bottom by the bottom's conjugate
你掌握了
- $z = x + iy$,$i^2 = -1$;共轭 $z^* = x - iy$;模 $|z| = \sqrt{x^2+y^2}$
- 画在阿尔冈图上;极坐标形式 $z = re^{i\theta}$
- 通过把分子分母同乘分母的共轭来除