复数fùshù

Core Concept

A Complex Number is an extension of real numbers, taking the form $z = a + bi$ where $a, b$ are real numbers and $i$ is the imaginary unit.

Imaginary Unit

The imaginary unit $i$ satisfies $i^2 = -1$.

$i^2 = -1, \quad i = \sqrt{-1}$

Powers of $i$:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = -i$
• $i^4 = 1$
• $i^{4k+r} = i^r$ ($k \in \mathbb{Z}, r \in \{0,1,2,3\}$)

Form of Complex Numbers

$z = a + bi$

Where:

• $a$ is the real part, denoted $\text{Re}(z)$
• $b$ is the imaginary part, denoted $\text{Im}(z)$
• When $b = 0$, $z$ is a real number
• When $a = 0, b \neq 0$, $z$ is a pure imaginary number
• When $b \neq 0$, $z$ is an imaginary number

Complex Equality

$a + bi = c + di \Leftrightarrow a = c \text{ and } b = d$

Complex Plane

Geometric Representation

Complex number $z = a + bi$ can be represented as point $(a, b)$ in the complex plane:

• Horizontal axis (real axis): Represents real part
• Vertical axis (imaginary axis): Represents imaginary part

Vector Representation

Complex number $z = a + bi$ can also be viewed as vector $\overrightarrow{OZ}$ from origin $O$ to point $(a, b)$.

Modulus of Complex Numbers

Definition

The modulus of complex number $z = a + bi$, denoted $|z|$:

$|z| = |a + bi| = \sqrt{a^2 + b^2}$

Geometric Meaning

$|z|$ represents the distance from point $z$ to the origin in the complex plane.

Properties

1. $|z| \geq 0$, with equality iff $z = 0$
2. $|z_1 \cdot z_2| = |z_1| \cdot |z_2|$
3. $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ ($z_2 \neq 0$)
4. $|z_1 + z_2| \leq |z_1| + |z_2|$ (triangle inequality)

Conjugate

Definition

The conjugate of complex number $z = a + bi$, denoted $\bar{z}$:

$\bar{z} = a - bi$

Geometric Meaning

$\bar{z}$ is the reflection of $z$ across the real axis.

Properties

1. $\overline{z_1 \pm z_2} = \bar{z_1} \pm \bar{z_2}$
2. $\overline{z_1 \cdot z_2} = \bar{z_1} \cdot \bar{z_2}$
3. $z \cdot \bar{z} = |z|^2 = a^2 + b^2$
4. $z + \bar{z} = 2a = 2\text{Re}(z)$
5. $z - \bar{z} = 2bi = 2i\text{Im}(z)$

Operations

Addition and Subtraction

$(a + bi) \pm (c + di) = (a \pm c) + (b \pm d)i$

Multiplication

$(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

Division

$\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$

Method: Multiply numerator and denominator by conjugate of denominator

CSCA Practice Problems

[Example 1] Basic (Difficulty ★★☆☆☆)

Given complex number $z = 3 + 4i$, find $|z|$ and $\bar{z}$.

Solution:

Modulus: $|z| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$

Conjugate: $\bar{z} = 3 - 4i$

Answer: $|z| = 5$, $\bar{z} = 3 - 4i$

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[Example 2] Intermediate (Difficulty ★★★☆☆)

Calculate $(2 + 3i)(1 - 2i)$.

Solution:

$(2 + 3i)(1 - 2i) = 2 - 4i + 3i - 6i^2$ $= 2 - i + 6 = 8 - i$

Answer: $8 - i$

Common Misconceptions

❌ Misconception 1: Treating $i$ as a variable

Wrong: Thinking $i$ can be further simplified like algebraic variables

Correct: $i$ is the imaginary unit with $i^2 = -1$, not a variable

❌ Misconception 2: Wrong modulus calculation

Wrong: $|3 + 4i| = 3 + 4 = 7$

Correct: $|3 + 4i| = \sqrt{3^2 + 4^2} = 5$

❌ Misconception 3: Wrong conjugate sign

Wrong: $\overline{3 + 4i} = -3 - 4i$

Correct: $\overline{3 + 4i} = 3 - 4i$ (only imaginary part changes sign)

Study Tips

1. ✅ Understand imaginary unit: $i^2 = -1$ is fundamental
2. ✅ Master operations: Addition, subtraction, multiplication, division
3. ✅ Remember modulus and conjugate: Their geometric meanings and properties
4. ✅ Practice division: Rationalizing denominator is key
5. ✅ Understand geometry: Points and vectors in complex plane

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💡 Exam Tip: Complex numbers are important in high school math. Relatively straightforward in CSCA exams, but basic operations and concepts must be mastered! Accounts for about 10-15% of algebra problems.