Complex Numbers — Definition, Formula & Examples

Q: What is the conjugate of a complex number and why is it useful?

The conjugate of $a + bi$ is $a - bi$ — you simply flip the sign of the imaginary part. When you multiply a complex number by its conjugate, the result is always a real number: $(a + bi)(a - bi) = a^2 + b^2$. This property is essential for dividing complex numbers, as shown in the division example above.

Complex Numbers

Numbers like 3 – 2i or The expression negative 1 plus the square root of 5 times i that can be written as the sum or difference of a real number and an imaginary number. Complex numbers are indicated by the symbol .

Note: All real numbers and all pure imaginary numbers are complex. Sometimes, however, mathematicians use the phrase complex numbers to refer strictly to numbers which have both nonzero real parts and nonzero imaginary parts.

Nested diagram showing number sets: natural, whole, integers, rationals, algebraic, reals, pure imaginary, and complex numbers...

Key Formula

z = a + bi

Where:

$z$ = The complex number
$a$ = The real part of the complex number (a real number)
$b$ = The imaginary part of the complex number (a real number)
$i$ = The imaginary unit, defined by i² = −1

Worked Example

Problem: Multiply the complex numbers (3 + 2i) and (1 − 4i).

Step 1: Apply the distributive property (FOIL) to expand the product.

(3 + 2i)(1 - 4i) = 3 \cdot 1 + 3 \cdot (-4i) + 2i \cdot 1 + 2i \cdot (-4i)

Step 2: Carry out each multiplication.

= 3 - 12i + 2i - 8i^2

Step 3: Replace

i^2

with

-1

= 3 - 12i + 2i - 8(-1) = 3 - 12i + 2i + 8

Step 4: Combine the real parts and the imaginary parts separately.

= (3 + 8) + (-12 + 2)i = 11 - 10i

Answer:

(3 + 2i)(1 - 4i) = 11 - 10i

Another Example

This example demonstrates division of complex numbers, which requires multiplying by the conjugate — a technique not needed in the first example's multiplication problem.

Problem: Divide the complex number (7 + i) by (2 − 3i).

Step 1: Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of

2 - 3i

2 + 3i

\frac{7 + i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i}

Step 2: Expand the numerator using FOIL.

(7 + i)(2 + 3i) = 14 + 21i + 2i + 3i^2 = 14 + 23i + 3(-1) = 11 + 23i

Step 3: Expand the denominator. A number times its conjugate gives

a^2 + b^2

(2 - 3i)(2 + 3i) = 4 + 9 = 13

Step 4: Write the result in standard form

a + bi

\frac{11 + 23i}{13} = \frac{11}{13} + \frac{23}{13}i

Answer:

\dfrac{7 + i}{2 - 3i} = \dfrac{11}{13} + \dfrac{23}{13}i

Frequently Asked Questions

What is the difference between real numbers and complex numbers?

Every real number is a complex number with its imaginary part equal to zero. For example, the real number

5

is the same as the complex number

5 + 0i

. Complex numbers extend real numbers by also allowing a nonzero imaginary part, such as

5 + 3i

. So the real numbers are a subset of the complex numbers.

What is the conjugate of a complex number and why is it useful?

The conjugate of

a + bi

a - bi

— you simply flip the sign of the imaginary part. When you multiply a complex number by its conjugate, the result is always a real number:

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

. This property is essential for dividing complex numbers, as shown in the division example above.

Can you graph complex numbers?

Yes. Complex numbers are plotted on the complex plane (also called the Argand plane), where the horizontal axis represents the real part and the vertical axis represents the imaginary part. For instance, the complex number

3 + 2i

is plotted at the point

(3, 2)

. This makes complex numbers behave like two-dimensional vectors.

Complex Numbers vs. Real Numbers

	Complex Numbers	Real Numbers
General form	$a + bi$ where $a, b \in \mathbb{R}$	Any point on the number line
Imaginary part	Can be nonzero	Always zero
Symbol	$\mathbb{C}$	$\mathbb{R}$
Graphed on	The complex plane (2D)	The number line (1D)
Ordering	Cannot be ordered (no "less than" or "greater than")	Fully ordered
Example	$3 - 2i$	$3$

Why It Matters

Complex numbers appear throughout algebra when you solve quadratic equations with negative discriminants — for example,

x^2 + 1 = 0

has no real solution, but it has the complex solutions

x = i

and

x = -i

. They are essential in advanced fields like electrical engineering (for analyzing alternating current circuits), quantum physics, and signal processing. In precalculus and calculus courses, you will use complex numbers to fully factor polynomials and apply the Fundamental Theorem of Algebra, which guarantees that every polynomial of degree

n

has exactly

n

roots in

\mathbb{C}

Common Mistakes

Mistake: Writing

i^2 = 1

instead of

i^2 = -1

Correction: The defining property of the imaginary unit is

i^2 = -1

. Forgetting the negative sign leads to wrong answers in every computation involving complex numbers. Always substitute

i^2 = -1

when simplifying.

Mistake: Treating the imaginary part of

a + bi

bi

instead of

b

Correction: The imaginary part is the real coefficient

b

, not

bi

. For example, the imaginary part of

3 + 7i

7

, not

7i

. This distinction matters when using formulas for the modulus or conjugate.

Related Terms

Imaginary Numbers — Numbers of the form bi, a subset of complex numbers
Real Numbers — Complex numbers with imaginary part zero
Real Part — The value a in the complex number a + bi
Imaginary Part — The value b in the complex number a + bi
Nonreal Numbers — Complex numbers with nonzero imaginary part
Gaussian Integer — Complex number where a and b are both integers
Rational Numbers — A subset of real numbers, hence also complex
Algebraic Numbers — Roots of polynomials, can be complex