Algebra and Geometry in the complex plane

The complex plane is the visualization of complex numbers as points in ${\mathbb{R}}^2$, with the real part on the horizontal axis and the imaginary part on the vertical axis. The geometric viewpoint connects complex algebra to plane geometry — addition becomes vector addition, multiplication becomes rotation and scaling.

Complex numbers

A complex number has the form

$z=x+iy$

where $x=\mathrm{Re}(z)$ is the real part, $y=\mathrm{Im}(z)$ is the imaginary part, and $i=\sqrt{-1}$.

The set of complex numbers is denoted $\mathbb{C}$. The real numbers form the subset where $\mathrm{Im}(z)=0$.

The complex plane is identified with ${\mathbb{R}}^2$ via the bijection $x+iy\leftrightarrow (x,y)$.

Addition and subtraction

Component-wise:

$(x_1+i{y}_1)+(x_2+i{y}_2)=(x_1+x_2)+i(y_1+y_2)$

Geometrically, this is parallelogram-rule vector addition.

Multiplication

Use $i^2=-1$ and distribute:

$(x_1+i{y}_1)(x_2+i{y}_2)=x_1{x}_2-y_1{y}_2+i(x_1{y}_2+x_2{y}_1)$

Geometrically, multiplication scales by the product of magnitudes and rotates by the sum of arguments — see Polar representation of complex numbers.

Conjugate

The complex conjugate of $z=x+iy$ is

$\bar{z}=x-iy$

Geometrically: reflection across the real axis. Properties:

• $z+\bar{z}=2\mathrm{Re}(z)$
• $z-\bar{z}=2i\mathrm{Im}(z)$
• $z\bar{z}=x^2+y^2=|z{|}^2$
• $\overline{z_1+z_2}={\bar{z}}_1+{\bar{z}}_2$
• $\overline{z_1{z}_2}={\bar{z}}_1{\bar{z}}_2$

Modulus (absolute value)

The modulus $|z|$ is the distance from the origin:

$|z|=\sqrt{x^2+y^2}$

Properties:

• $|z|\ge 0$, with equality iff $z=0$.
• $|z_1{z}_2|=|z_1|\cdot |z_2|$.
• $|z_1+z_2|\le |z_1|+|z_2|$ (triangle inequality).

Division

Multiply numerator and denominator by the conjugate of the denominator:

$\frac{z_1}{z_2}=\frac{z_1{\bar{z}}_2}{z_2{\bar{z}}_2}=\frac{z_1{\bar{z}}_2}{|z_2{|}^2}$

This rationalizes the denominator and gives an explicit form for the quotient.

Algebraic structure

$\mathbb{C}$ forms a field: closed under addition, multiplication, and division (by nonzero elements), with associative, commutative, and distributive laws. It’s algebraically closed — every non-constant polynomial with complex coefficients has a complex root (Fundamental Theorem of Algebra).

Why this matters

Complex numbers come up in:

• Electrical engineering: phasors and AC circuit analysis. See Phasor.
• Signal processing: Fourier transforms.
• Quantum mechanics: wave functions are complex-valued.
• Differential equations: complex eigenvalues correspond to oscillatory solutions. See Complex conjugate eigenvalues case.
• Geometry: complex multiplication is rotation + scaling.

For the polar form (radius/angle parameterization), see Polar representation of complex numbers.