4.1 Algebraic and Geometric Representations of Complex Numbers

4.1 Algebraic and Geometric Representations of Complex Numbers

The expansion of the real number system $\mathbb{R}$ to the complex number system $\mathbb{C}$ represents one of the most profound developments in mathematics. In the real number field $\mathbb{R}$, equations such as $x^2 + 1 = 0$ have no solutions because the square of any real number is non-negative. By introducing the imaginary unit $i$, defined by $i^2 = -1$, mathematicians constructed the complex number system $\mathbb{C}$, providing solutions to all polynomial equations and establishing a deep bridge between algebra, geometry, and trigonometry.

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1. Algebraic Definitions and Classifications

A complex number is an expression of the form:

$$z = a + bi \quad (a, b \in \mathbb{R})$$

where $a$ is called the real part of $z$, denoted as $\operatorname{Re}(z) = a$, and $b$ is called the imaginary part of $z$, denoted as $\operatorname{Im}(z) = b$. Note that both the real part and the imaginary part are real numbers themselves.

Classification of Complex Numbers

The set of all complex numbers is denoted by $\mathbb{C}$. Based on the values of $a$ and $b$, complex numbers are classified as follows:

• Real Numbers ($\mathbb{R}$): When $b = 0$, $z = a \in \mathbb{R}$. Thus, $\mathbb{R} \subset \mathbb{C}$.
• Imaginary Numbers: When $b \neq 0$, $z = a + bi$ is an imaginary number.
• Purely Imaginary Numbers: When $a = 0$ and $b \neq 0$, $z = bi$ is called a purely imaginary number.

Equality of Complex Numbers

Two complex numbers $z_1 = a + bi$ and $z_2 = c + di$ ($a, b, c, d \in \mathbb{R}$) are equal if and only if their real parts are equal and their imaginary parts are equal:

$$z_1 = z_2 \iff a = c \quad \text{and} \quad b = d$$

Gaokao Strategy Note: Inequality relationships ($z_1 > z_2$ or $z_1 < z_2$) are generally not defined for non-real complex numbers. If an inequality $z_1 > z_2$ is stated in a Gaokao question, it implicitly implies that both $z_1$ and $z_2$ must be real numbers.

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2. Geometric Representation: The Complex Plane

Every complex number $z = a + bi$ can be uniquely paired with an ordered pair of real numbers $(a, b)$. This establishes a natural bijection between the set of complex numbers $\mathbb{C}$ and points in the two-dimensional Cartesian plane:

$$\mathbb{C} \longleftrightarrow \mathbb{R}^2 \quad \text{via} \quad z = a + bi \longleftrightarrow P(a, b)$$

When the Cartesian coordinate plane is used to represent complex numbers, it is called the complex plane (or Argand diagram):

• The horizontal axis is called the real axis (representing $\operatorname{Re}(z)$).
• The vertical axis is called the imaginary axis (representing $\operatorname{Im}(z)$), excluding the origin for purely imaginary numbers.

Vector Interpretation

Furthermore, there is a one-to-one correspondence between complex numbers $z = a + bi$ and position vectors starting from the origin:

$$z = a + bi \longleftrightarrow \vec{OP} = (a, b)$$

This vector interpretation allows complex addition and subtraction to be understood as vector addition and subtraction obeying the parallelogram rule.

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3. Modulus and Complex Conjugate

Modulus of a Complex Number

The modulus (or absolute value) of a complex number $z = a + bi$, denoted by $|z|$, is defined as the Euclidean distance from the origin $O(0,0)$ to the point $P(a,b)$ in the complex plane:

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

Fundamental Properties of Modulus:

1. Non-negativity: $|z| \geq 0$, with $|z| = 0 \iff z = 0$.
2. Multiplicativity: $|z_1 z_2| = |z_1| \cdot |z_2|$ and $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ (for $z_2 \neq 0$).
3. Triangle Inequality: $|z_1 + z_2| \leq |z_1| + |z_2|$, with equality holding if and only if $\vec{OP_1}$ and $\vec{OP_2}$ are collinear and point in the same direction.
4. Distance Formula: The distance between two complex numbers $z_1 = a_1 + b_1 i$ and $z_2 = a_2 + b_2 i$ in the complex plane is given by $|z_1 - z_2| = \sqrt{(a_1 - a_2)^2 + (b_1 - b_2)^2}$.

Complex Conjugate

The complex conjugate of $z = a + bi$, denoted by $\bar{z}$ or $z^*$, is defined as:

$$\bar{z} = a - bi$$

Geometrically, $\bar{z}$ represents the reflection of $z$ across the real axis.

Key Algebraic Properties of Conjugates:

• $\overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2$
• $\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2$ and $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z}_1}{\bar{z}_2}$
• $z \bar{z} = (a + bi)(a - bi) = a^2 + b^2 = |z|^2$
• Real and Imaginary Part Extraction: $\operatorname{Re}(z) = \frac{z + \bar{z}}{2}$ and $\operatorname{Im}(z) = \frac{z - \bar{z}}{2i}$
• $z$ is a real number $\iff z = \bar{z}$; $z$ is a purely imaginary number $\iff z = -\bar{z} \neq 0$.

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4. Trigonometric and Exponential (Euler) Forms

By introducing polar coordinates $(r, \theta)$ in the complex plane where $x = r\cos\theta$ and $y = r\sin\theta$:

$$r = |z| = \sqrt{a^2 + b^2}, \quad \cos\theta = \frac{a}{r}, \quad \sin\theta = \frac{b}{r} \quad (r > 0)$$

We can express $z$ in trigonometric form:

$$z = r(\cos\theta + i\sin\theta)$$

where $r$ is the modulus and $\theta$ is an argument of $z$, written as $\theta = \arg(z)$. The argument is multi-valued ($\theta + 2k\pi, k \in \mathbb{Z}$). The unique value of $\theta$ in the interval $(-\pi, \pi]$ (or $[0, 2\pi)$) is called the principal argument, denoted by $\operatorname{Arg}(z)$.

Euler's Formula

Using Euler's formula, $e^{i\theta} = \cos\theta + i\sin\theta$, any complex number can be written in exponential form:

$$z = r e^{i\theta}$$

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5. Worked Gaokao Exam Examples

Example 1: Purely Imaginary Condition and Parameter Determination

Problem: Let $m \in \mathbb{R}$. If the complex number $z = (m^2 - 3m + 2) + (m^2 - 1)i$ is a purely imaginary number, find the value of $m$ and calculate $|z|$.

Solution:

1. For $z$ to be a purely imaginary number, its real part must equal zero while its imaginary part must be non-zero: $$\begin{cases} \operatorname{Re}(z) = m^2 - 3m + 2 = 0 \\ \operatorname{Im}(z) = m^2 - 1 \neq 0 \end{cases}$$
2. Solve the quadratic equation for the real part: $$m^2 - 3m + 2 = (m - 1)(m - 2) = 0 \implies m = 1 \text{ or } m = 2$$
3. Check the non-zero condition for the imaginary part:
   • If $m = 1$, $\operatorname{Im}(z) = 1^2 - 1 = 0$, which makes $z = 0 + 0i = 0$ (a real number, not purely imaginary). Thus $m = 1$ is rejected.
   • If $m = 2$, $\operatorname{Im}(z) = 2^2 - 1 = 3 \neq 0$.
4. Thus, $m = 2$. Substituting $m = 2$ into $z$ gives $z = 0 + 3i = 3i$.
5. The modulus is $|z| = |3i| = \sqrt{0^2 + 3^2} = 3$.

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Example 2: Complex Locus and Distance Analysis

Problem: In the complex plane, let $z$ satisfy the equation $|z - (1 + 2i)| = \sqrt{5}$. Describe the geometric locus of $z$ and find the maximum value of $|z|$.

Solution:

1. Let $z = x + yi$ ($x, y \in \mathbb{R}$). The given equation becomes: $$|(x - 1) + (y - 2)i| = \sqrt{5} \implies \sqrt{(x - 1)^2 + (y - 2)^2} = \sqrt{5}$$ Squaring both sides yields $(x - 1)^2 + (y - 2)^2 = 5$.
2. Geometrically, this equation represents a circle centered at $C(1, 2)$ with radius $R = \sqrt{5}$.
3. $|z|$ represents the distance from the origin $O(0,0)$ to the point $P(x,y)$ on the circle.
4. The distance from the origin $O(0,0)$ to the center $C(1,2)$ is: $$|OC| = \sqrt{1^2 + 2^2} = \sqrt{5}$$
5. Since the center is at distance $\sqrt{5}$ from the origin and the radius is $R = \sqrt{5}$, the circle passes through the origin $O(0,0)$.
6. The maximum distance $|z|_{\max}$ occurs at the point diametrically opposite to the origin: $$|z|_{\max} = |OC| + R = \sqrt{5} + \sqrt{5} = 2\sqrt{5}$$

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6. Summary Comparison of Complex Representations