Easy

Complex Numbers in Standard Form

Simple Explanation

A complex number combines a real part and an imaginary part into one number, written in standard form z = a + bi, where a and b are ordinary real numbers and i is the imaginary unit.

Why Do We Need It?

Standard form is the universal way complex numbers are written — every operation (addition, multiplication, division) in this chapter is defined directly in terms of this a+bi structure.

Formula

Standard Form of a Complex Number

z = a + bi

Every complex number can be written as a real number a plus a real number b times i — a is the real part, b is the imaginary part.

a: — the real part of z, written Re(z)
b: — the imaginary part of z, written Im(z) — a real number, the coefficient of i

When to use it: Whenever a complex number needs to be identified or written in its most basic, universal form.

Worked Example

Identify the real and imaginary parts

Identify the real part and imaginary part of z = 5 − 3i.

Why Does This Work?

Defining Im(z) as a plain real number (the coefficient of i, not the term "bi" itself) keeps it consistent with Re(z) — both parts of a complex number are ordinary real numbers, which is what lets a+bi be plotted as an ordinary point (a,b) on a plane later in this chapter.

Real-Life Example

Representing electrical impedance

An engineer describes a circuit component's impedance using both a resistive part and a reactive part.

These two physically different quantities are naturally combined into one complex number in standard form, with resistance as the real part and reactance as the imaginary part.

Practice

For z = −7 + 2i, find Im(z).

Easy

Common mistake

Writing Im(z) as "2i" instead of just "2" — the imaginary part is defined as the real coefficient of i, not the whole term including i.

Quick Review

z = a + bi is standard form, with a and b both real numbers.
Re(z) = a is the real part.
Im(z) = b is the imaginary part — a real number, the coefficient of i.