Algebra and Geometry in the complex plane

The complex plane plots complex numbers as points in ${\mathbb{R}}^2$: real part on the horizontal axis, imaginary part on the vertical. The geometric view 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:

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

The conjugate clears the imaginary part out of the denominator, leaving real and imaginary parts in standard form.

Algebraic structure

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

Where this shows up

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