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\( a+bi \)
 C

Operations with
Complex Numbers

Master addition, subtraction, and multiplication of complex numbers using the relation \( i^2 = -1 \) and familiar algebraic properties.

Algebra 2

What Is a Complex Number?

A complex number has the standard form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. The symbol \( i \) represents the imaginary unit, defined by the property \( i^2 = -1 \).

Real Part \( a \)
The ordinary number on the horizontal axis: e.g., in \( 3 + 5i \), the real part is \( 3 \).
Imaginary Part \( b \)
The coefficient of \( i \) on the vertical axis: e.g., in \( 3 + 5i \), the imaginary part is \( 5 \).

Visualizing on the Complex Plane

\(\mathrm{Re}\)
\(\mathrm{Im}\)
 -2 -1 1 2 3
\(2i\)
\(i\)
\(-i\)
\(-2i\)
\( 3+2i \)
\( -1-i \)

Why Do Complex Numbers Matter?

Real numbers alone can't solve every equation. The equation \( x^2 + 1 = 0 \) has no real solution, but it has two complex solutions: \( x = i \) and \( x = -i \). Complex numbers complete the number system so every polynomial has roots.

Algebra
Every polynomial of degree \( n \) has exactly \( n \) roots (counting multiplicity) in the complex numbers.
Engineering
Complex numbers model alternating currents, signal processing, and control systems.
Physics
Quantum mechanics uses complex-valued wave functions as a fundamental tool.

The Core Rule: \( i^2 = -1 \)

The defining property of \( i \) is that \( i^2 = -1 \). This single relation, combined with the commutative, associative, and distributive properties, is all you need to add, subtract, and multiply complex numbers.

\( i^1 \)
\( = i \)
\( i^2 \)
\( = -1 \)
\( i^3 \)
\( = -i \)
\( i^4 \)
\( = 1 \)

Properties You Already Know

Complex number arithmetic uses the same algebraic properties as real numbers. You don't need to learn new rules: just apply what you know and simplify with \( i^2 = -1 \).

1
Commutative: \( z_1 + z_2 = z_2 + z_1 \) and \( z_1 \cdot z_2 = z_2 \cdot z_1 \)
2
Associative: \( (z_1 + z_2) + z_3 = z_1 + (z_2 + z_3) \)
3
Distributive: \( z_1(z_2 + z_3) = z_1 z_2 + z_1 z_3 \)

Adding and Subtracting

To add or subtract complex numbers, combine real parts and combine imaginary parts separately:

\(\mathrm{Re}\)
\(\mathrm{Im}\)
 1 2 3 4 1 2 3
\( 3+2i \)
\( 1+i \)
\( 4+3i \)

Multiplying Complex Numbers

Multiply complex numbers using FOIL (or the distributive property), then replace \( i^2 \) with \( -1 \):

The result is always in standard form \( a + bi \): a real part \( (ac - bd) \) and an imaginary part \( (ad + bc) \).

Real vs. Imaginary vs. Complex

Type Form Example Note
Real \( a + 0i \) \( 5 \) \( b = 0 \)
Pure Imaginary \( 0 + bi \) \( 3i \) \( a = 0, \; b \neq 0 \)
Complex \( a + bi \) \( 2 + 7i \) Neither real nor pure imaginary

How to Apply the Operations

1
Identify the operation: Are you adding, subtracting, or multiplying?
2
For addition/subtraction: Group real parts together and imaginary parts together, then combine.
3
For multiplication: Use the distributive property (FOIL). Expand all four products.
4
Simplify: Replace every \( i^2 \) with \( -1 \).
5
Write in standard form: Combine and express as \( a + bi \).

Example 1: Addition

Simplify \( (4 + 3i) + (2 - 5i) \).

1
Group real parts: \( 4 + 2 = \) \( 6 \)
2
Group imaginary parts: \( 3i + (-5i) = 3i - 5i = \) \( -2i \)
3
Combine: \( (4 + 3i) + (2 - 5i) = \) \( 6 - 2i \)

Example 2: Subtraction

Simplify \( (7 - 2i) - (3 + 4i) \).

1
Distribute the negative: \( 7 - 2i - 3 - 4i \)
2
Real parts: \( 7 - 3 = \) \( 4 \)
3
Imaginary parts: \( -2i - 4i = \) \( -6i \)
4
Result: \( 4 - 6i \)

Example 3: Multiplication

Simplify \( (3 + 2i)(1 - 4i) \).

First
\( 3 \cdot 1 = 3 \)
 Outer
\( 3(-4i)=-12i \)
 Inner
\( 2i \cdot 1 = 2i \)
 Last
\( 2i(-4i)=-8i^2 \)
\( 3 - 12i + 2i - 8i^2 = 3 - 10i -8(-1) = 11 - 10i \)

Example 4: Squaring a Complex Number

Simplify \( (2 + 3i)^2 \).

1
Expand: \( (2 + 3i)^2 = (2 + 3i)(2 + 3i) \)
2
FOIL: \( 4 + 6i + 6i + 9i^2 \)
3
Replace \( i^2 = -1 \): \( 4 + 12i + 9(-1) = 4 + 12i - 9 \)
4
Simplify: \( -5 + 12i \)

Common Mistakes to Avoid

✕ Forgetting \( i^2 = -1 \)
Leaving an answer as \( 6i^2 \) instead of simplifying it to \( -6 \). Always replace \( i^2 \) with \( -1 \).
✕ Dropping the negative sign
Computing \( (5-2i)-(3+i) \) as \( 2 - i \) instead of \( 2 - 3i \). Distribute the minus to both terms.
✕ Squaring term-by-term
Writing \( (a+bi)^2 = a^2 + b^2 i^2 \). You must FOIL: \( (a+bi)^2 = a^2 + 2abi + b^2 i^2 \).

Operation Summary

Operation Formula Example
Addition \( (a+c) + (b+d)i \) \( (2+3i)+(1+4i) = 3+7i \)
Subtraction \( (a-c) + (b-d)i \) \( (5+i)-(2+3i) = 3-2i \)
Multiplication \( (ac-bd)+(ad+bc)i \) \( (1+2i)(3+i) = 1+7i \)

Question 1 of 5

Simplify \( (6 + 2i) + (3 - 5i) \).

Question 2 of 5

Simplify \( (8 - 3i) - (2 + 4i) \).

Question 3 of 5

Simplify \( (2 + 5i)(3 - i) \).

Question 4 of 5

What is \( (1 + i)^2 \) in standard form?

Question 5 of 5

Simplify \( 4i \cdot 3i \).