Number Systems

What Are Complex Numbers?

A complex number has the form a+bi, where a and b are real and i^2=-1. Real numbers are the special case b=0.

Reviewed 2026-07-26

Real and imaginary components

The real part of z=a+bi is a and the imaginary part is b. Two complex numbers are equal only when both corresponding parts are equal.

Complex numbers form a plane rather than a line: the horizontal axis represents real parts and the vertical axis represents imaginary parts.

Formulaz=a+bi; Re(z)=a; Im(z)=b

Arithmetic and conjugates

Addition combines matching components. Multiplication distributes normally and replaces i^2 with -1. The conjugate a-bi reflects a point across the real axis.

Multiplying a complex number by its conjugate gives the real value a^2+b^2, which makes complex division possible.

Formula(a+bi)(a-bi)=a^2+b^2

Modulus and polar form

The modulus |z| is the distance from the origin. The argument is an angle locating the number's direction, with quadrant handled by an atan2-type function.

Polar form separates magnitude and angle, making multiplication and division geometric: magnitudes multiply or divide while arguments add or subtract.

Formula|z|=sqrt(a^2+b^2); z=r(cos(theta)+i sin(theta))

Where this idea is used

Solving polynomial equations with no real roots.
Representing electrical phase, waves, and rotations.
Simplifying two-dimensional scaling and rotation.

Worked examples

Multiply (2+3i)(4-i)

Distribute and combine real and imaginary terms.

8-2i+12i-3i^2
i^2=-1, so -3i^2=3
real=11, imaginary=10i

Result: 11+10i

Find the modulus of -3+4i

Use the distance formula.

|-3+4i|=sqrt((-3)^2+4^2)
sqrt(9+16)=sqrt25

Result: |-3+4i|=5

Common mistakes

Treating i^2 as 1 instead of -1.
Using ordinary arctangent without correcting the quadrant.
Dividing real and imaginary parts separately instead of multiplying by the denominator's conjugate.

Questions about complex numbers

Are imaginary numbers less real than real numbers?

They are well-defined mathematical numbers; the name is historical. Complex arithmetic is internally consistent and widely applied.

Is every real number complex?

Yes. A real number a can be written a+0i.

What is the argument of zero?

It is undefined because the origin has no unique direction.