The Complex Number Field & Geometry of the Complex Plane

1. Algebraic Construction of $\mathbb{C}$

The set of real numbers $\mathbb{R}$ is algebraically incomplete: polynomial equations such as $x^2 + 1 = 0$ possess no real roots. We construct the field of complex numbers $\mathbb{C}$ as the set of ordered pairs of real numbers: $$\mathbb{C} \equiv \{(x, y) \in \mathbb{R}^2\}$$ equipped with two binary operations, addition ($+$) and multiplication ($\cdot$):

2. Verification of Field Axioms

The algebraic structure $(\mathbb{C}, +, \cdot)$ satisfies all nine field axioms:

• Additive Identity: $0_{\mathbb{C}} = (0, 0)$. For any $z = (x, y)$, $z + 0_{\mathbb{C}} = (x+0, y+0) = z$.
• Additive Inverse: For each $z = (x, y)$, $-z = (-x, -y)$, yielding $z + (-z) = (0, 0)$.
• Multiplicative Identity: $1_{\mathbb{C}} = (1, 0)$. For any $z = (x, y)$, $(x, y)(1, 0) = (x\cdot 1 - y\cdot 0, x\cdot 0 + y\cdot 1) = (x, y)$.
• Multiplicative Inverse: For every non-zero $z = (x, y) \ne (0, 0)$, $x^2 + y^2 > 0$. The inverse is: $$z^{-1} = \left(\frac{x}{x^2 + y^2}, \; \frac{-y}{x^2 + y^2}\right)$$ Direct calculation yields $z \cdot z^{-1} = \left(\frac{x^2 + y^2}{x^2 + y^2}, \; \frac{-xy + yx}{x^2 + y^2}\right) = (1, 0)$.
• Distributivity & Commutativity: Multiplication distributes over addition, and both operations are commutative and associative.

3. Canonical Form and the Imaginary Unit

The map $\iota: \mathbb{R} \hookrightarrow \mathbb{C}$ defined by $\iota(x) = (x, 0)$ is an injective field homomorphism, allowing us to identify $\mathbb{R}$ as a subfield of $\mathbb{C}$. Defining the imaginary unit $i \equiv (0, 1)$: $$i^2 = (0, 1) \cdot (0, 1) = (0\cdot 0 - 1\cdot 1, \; 0\cdot 1 + 1\cdot 0) = (-1, 0) \equiv -1$$ Any element $z = (x, y)$ can be written in the canonical Cartesian form: $$\mathbf{z = (x, 0) + (0, y) = x(1, 0) + y(0, 1) = x + iy}$$ where $x = \text{Re}(z) \in \mathbb{R}$ is the real part and $y = \text{Im}(z) \in \mathbb{R}$ is the imaginary part.

4. Non-Orderability of $\mathbb{C}$

Theorem: The complex field $\mathbb{C}$ cannot be endowed with the structure of an ordered field.
Proof: In any ordered field, squares of non-zero elements are strictly positive: $a \ne 0 \implies a^2 > 0$. Consequently, $1^2 = 1 > 0$, which implies $-1 < 0$. If $\mathbb{C}$ were ordered, $i \ne 0$ would force $i^2 > 0 \implies -1 > 0$, contradicting $-1 < 0$. Hence, no compatible total order exists on $\mathbb{C}$. $\blacksquare$

1. The Argand Representation

Jean-Robert Argand and Carl Friedrich Gauss introduced the geometric representation of $\mathbb{C}$ as a two-dimensional Euclidean plane $\mathbb{R}^2$: the horizontal axis represents the real axis ($\text{Re}$), and the vertical axis represents the imaginary axis ($\text{Im}$). Every complex number $z = x + iy$ corresponds to the unique point $P(x, y)$ or the position vector $\vec{OP}$.

2. The Complex Conjugate

For $z = x + iy \in \mathbb{C}$, its complex conjugate $\bar{z}$ is defined by: $$\mathbf{\bar{z} \equiv x - iy}$$ Geometrically, $\bar{z}$ is the orthogonal reflection of $z$ across the real axis. Fundamental Properties of Conjugation:

• $\overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2$ and $\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2$.
• $\overline{(z_1 / z_2)} = \bar{z}_1 / \bar{z}_2$ for $z_2 \ne 0$.
• $\overline{\bar{z}} = z$.
• $\text{Re}(z) = \frac{z + \bar{z}}{2}, \quad \text{Im}(z) = \frac{z - \bar{z}}{2i}$.
• $z \in \mathbb{R} \iff z = \bar{z}; \quad z \text{ is purely imaginary} \iff z = -\bar{z}$.

3. The Modulus (Absolute Value)

The modulus of $z = x + iy$, denoted $|z|$, is the Euclidean distance from the origin to $(x, y)$: $$\mathbf{|z| \equiv \sqrt{x^2 + y^2} = \sqrt{z \bar{z}}}$$ Key algebraic properties:

• $|z| \ge 0$, and $|z| = 0 \iff z = 0$.
• $|z| = |\bar{z}| = |-z| = |-\bar{z}|$.
• $z \bar{z} = |z|^2$. Thus, division can be computed algebraically as $\frac{w}{z} = \frac{w \bar{z}}{|z|^2}$.
• $|z_1 z_2| = |z_1| |z_2|$. Proof: $|z_1 z_2|^2 = (z_1 z_2)\overline{(z_1 z_2)} = z_1 z_2 \bar{z}_1 \bar{z}_2 = (z_1 \bar{z}_1)(z_2 \bar{z}_2) = |z_1|^2 |z_2|^2$. Taking non-negative square roots gives $|z_1 z_2| = |z_1||z_2|$.
• $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ for $z_2 \ne 0$.

1. The Fundamental Triangle Inequality

Theorem: For any two complex numbers $z_1, z_2 \in \mathbb{C}$: $$\mathbf{|z_1 + z_2| \le |z_1| + |z_2|}$$ Rigorous Proof: Expanding the squared modulus: $$|z_1 + z_2|^2 = (z_1 + z_2)\overline{(z_1 + z_2)} = (z_1 + z_2)(\bar{z}_1 + \bar{z}_2) = z_1 \bar{z}_1 + z_1 \bar{z}_2 + z_2 \bar{z}_1 + z_2 \bar{z}_2$$ Since $z_2 \bar{z}_1 = \overline{z_1 \bar{z}_2}$, their sum is $2\text{Re}(z_1 \bar{z}_2)$: $$|z_1 + z_2|^2 = |z_1|^2 + 2\text{Re}(z_1 \bar{z}_2) + |z_2|^2$$ For any complex number $w$, $\text{Re}(w) \le |w|$. Therefore: $$\text{Re}(z_1 \bar{z}_2) \le |z_1 \bar{z}_2| = |z_1||\bar{z}_2| = |z_1||z_2|$$ Substituting this upper bound: $$|z_1 + z_2|^2 \le |z_1|^2 + 2|z_1||z_2| + |z_2|^2 = (|z_1| + |z_2|)^2$$ Taking the positive square root on both sides yields: $$|z_1 + z_2| \le |z_1| + |z_2| \quad \blacksquare$$

2. Condition for Equality

Equality holds if and only if $\text{Re}(z_1 \bar{z}_2) = |z_1 \bar{z}_2|$, which requires $z_1 \bar{z}_2$ to be a non-negative real number. If $z_2 \ne 0$, this means: $$\frac{z_1}{z_2} = \frac{z_1 \bar{z}_2}{|z_2|^2} = \lambda \ge 0$$ Geometrically, equality holds if and only if the vectors $z_1$ and $z_2$ lie on the same ray emanating from the origin (same direction).

3. The Reverse Triangle Inequality

Theorem: For any $z_1, z_2 \in \mathbb{C}$: $$\mathbf{||z_1| - |z_2|| \le |z_1 - z_2|}$$ Proof: Write $z_1 = (z_1 - z_2) + z_2$. By the triangle inequality: $$|z_1| = |(z_1 - z_2) + z_2| \le |z_1 - z_2| + |z_2| \implies |z_1| - |z_2| \le |z_1 - z_2|$$ Similarly, interchanging $z_1$ and $z_2$: $$|z_2| - |z_1| \le |z_2 - z_1| = |z_1 - z_2| \implies -(|z_1| - |z_2|) \le |z_1 - z_2|$$ Combining these two inequalities gives $||z_1| - |z_2|| \le |z_1 - z_2|$. $\blacksquare$

Combining both results yields the complete bounds: $$\mathbf{||z_1| - |z_2|| \le |z_1 \pm z_2| \le |z_1| + |z_2|}$$

1. Modulus-Argument (Polar) Representation

Let $z = x + iy \ne 0$. Introducing plane polar coordinates $x = r\cos\theta$ and $y = r\sin\theta$, where $r = |z| = \sqrt{x^2 + y^2} > 0$: $$\mathbf{z = r(\cos\theta + i\sin\theta)}$$ The angle $\theta$ is the argument of $z$, denoted $\arg(z)$. Because $\sin$ and $\cos$ are $2\pi$-periodic, the argument is multi-valued: $$\arg(z) = \text{Arg}(z) + 2k\pi, \quad k \in \mathbb{Z}$$ The principal argument $\text{Arg}(z)$ is uniquely chosen in the interval $(-\pi, \pi]$: $$\text{Arg}(z) = \begin{cases} \arctan(y/x) & x > 0 \\ \arctan(y/x) + \pi & x < 0, \; y \ge 0 \\ \arctan(y/x) - \pi & x < 0, \; y < 0 \\ \pi/2 & x = 0, \; y > 0 \\ -\pi/2 & x = 0, \; y < 0 \end{cases}$$

2. Euler's Formula

By expanding the complex exponential function via its Taylor series: $$e^{i\theta} = \sum_{n=0}^\infty \frac{(i\theta)^n}{n!} = \sum_{k=0}^\infty \frac{(-1)^k \theta^{2k}}{(2k)!} + i \sum_{k=0}^\infty \frac{(-1)^k \theta^{2k+1}}{(2k+1)!} = \cos\theta + i\sin\theta$$ This gives Euler's celebrated representation: $$\mathbf{z = r e^{i\theta}}$$

3. Multiplication and Division in Polar Form

Let $z_1 = r_1 e^{i\theta_1}$ and $z_2 = r_2 e^{i\theta_2}$. Then: $$z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} = r_1 r_2 [\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$$ $$\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$$ Geometric Interpretation: Multiplying $z_1$ by $z_2$ scales the magnitude of $z_1$ by $r_2$ and rotates the vector counterclockwise by angle $\theta_2$. In particular, multiplying any complex number by $i = e^{i\pi/2}$ performs an exact $90^{\circ}$ counterclockwise rotation.

1. Equation of a Straight Line in $\mathbb{C}$

A line in the Cartesian plane has the equation $Ax + By + C = 0$ ($A, B, C \in \mathbb{R}$). Substituting $x = \frac{z + \bar{z}}{2}$ and $y = \frac{z - \bar{z}}{2i}$: $$A\left(\frac{z + \bar{z}}{2}\right) + B\left(\frac{z - \bar{z}}{2i}\right) + C = 0 \iff \left(\frac{A - iB}{2}\right)z + \left(\frac{A + iB}{2}\right)\bar{z} + C = 0$$ Defining the complex coefficient $\alpha \equiv \frac{A + iB}{2} \in \mathbb{C}$ and real constant $\beta \equiv C \in \mathbb{R}$: $$\mathbf{\bar{\alpha} z + \alpha \bar{z} + \beta = 0}$$

2. Equation of a Circle in $\mathbb{C}$

A circle centered at $z_0 \in \mathbb{C}$ with radius $R > 0$ is the locus $|z - z_0| = R$. Squaring both sides: $$(z - z_0)\overline{(z - z_0)} = R^2 \iff z \bar{z} - \bar{z}_0 z - z_0 \bar{z} + (|z_0|^2 - R^2) = 0$$ The general circle equation in $\mathbb{C}$ is: $$\mathbf{z \bar{z} + \bar{\alpha} z + \alpha \bar{z} + k = 0, \quad k \in \mathbb{R}, \; |\alpha|^2 - k > 0}$$ with center $z_c = -\alpha$ and radius $R = \sqrt{|\alpha|^2 - k}$.

3. Apollonian Circles in the Complex Plane

Theorem: For two distinct points $z_1, z_2 \in \mathbb{C}$ and a constant $k > 0$, the locus of points satisfying: $$\left|\frac{z - z_1}{z - z_2}\right| = k$$ is:

• For $k = 1$: The perpendicular bisector of the segment connecting $z_1$ and $z_2$.
• For $k \ne 1$: A circle (termed the Apollonian circle) with center $z_c$ and radius $R$: $$\mathbf{z_c = \frac{z_1 - k^2 z_2}{1 - k^2}, \qquad R = \frac{k |z_1 - z_2|}{|1 - k^2|}}$$