Algebra 247 views·Updated Aug 13, 2026·3 pages

Mastering Operations with Complex Numbers: Add, Subtract, Multiply & Divide

Ahmed AK.@ahmed_ak.

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Complex numbers blend real and imaginary parts, written as a...

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Adding, Subtracting, Multiplying and Dividing Complex Numbers – page 1

Understanding Complex Numbers

Complex numbers have two parts: a real number and an imaginary number written as a + bi. The mysterious i is defined as the square root of -1, something you can't find on a regular number line!

Working with complex numbers isn't as scary as it sounds. When adding or subtracting, you simply combine like terms - real parts with real parts, imaginary parts with imaginary parts.

For example, to add $5+2i$ + $-5+4i$ , you combine the real parts $5-5$ and the imaginary parts $2i+4i$ to get 0+6i, or simply 6i.

Quick Tip: Think of complex numbers like combining apples and oranges - you can't mix them directly, but you can count how many of each you have!

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Adding, Subtracting, Multiplying and Dividing Complex Numbers – page 2

The Powers of i

When working with complex numbers, understanding the pattern of i raised to different powers is super helpful. The pattern cycles every four powers!

When i is squared (i²), we get -1, because i × i = √-1 × √-1 = -1. This is the foundation for all other powers. For i³, we multiply i² × i, getting -1 × i = -i. Then i⁴ = i² × i² = $-1$$-1$ = 1.

The pattern continues: i⁵ = i, i⁶ = -1, i⁷ = -i, i⁸ = 1, and so on. This creates a repeating cycle of four values: i, -1, -i, 1.

Remember This: The powers of i follow a simple pattern: i, -1, -i, 1. If you need to find a higher power, divide the exponent by 4 and look at the remainder!

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Adding, Subtracting, Multiplying and Dividing Complex Numbers – page 3

Multiplying Complex Numbers

Multiplying complex numbers requires applying the pattern of i powers. When you multiply expressions with i, remember that i² equals -1.

Let's try an example: (3) $2-7i$ + (4i) $2-7i$ . First, multiply 3 by each term in $2-7i$ to get 6-21i. Then multiply 4i by each term in $2-7i$ to get 8i-28i².

Since i² equals -1, we can rewrite 28i² as 28 $-1$ or -28. Combining all terms: 6-21i+8i+28 = 34-13i.

Pro Tip: When multiplying complex numbers, distribute first just like with regular algebraic expressions, then substitute i² = -1, i³ = -i, and i⁴ = 1 wherever needed!

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Algebra 247 views·Updated Aug 13, 2026·3 pages

Mastering Operations with Complex Numbers: Add, Subtract, Multiply & Divide

Ahmed AK.@ahmed_ak.

Add to Google sources

Complex numbers blend real and imaginary parts, written as a + bi, where i = √-1. They're a powerful mathematical tool that extends beyond real numbers, allowing us to solve equations that previously seemed impossible.

of 3

Understanding Complex Numbers

Complex numbers have two parts: a real number and an imaginary number written as a + bi. The mysterious i is defined as the square root of -1, something you can't find on a regular number line!

Working with complex numbers isn't as scary as it sounds. When adding or subtracting, you simply combine like terms - real parts with real parts, imaginary parts with imaginary parts.

For example, to add $5+2i$ + $-5+4i$ , you combine the real parts $5-5$ and the imaginary parts $2i+4i$ to get 0+6i, or simply 6i.

Quick Tip: Think of complex numbers like combining apples and oranges - you can't mix them directly, but you can count how many of each you have!

of 3

The Powers of i

When working with complex numbers, understanding the pattern of i raised to different powers is super helpful. The pattern cycles every four powers!

The pattern continues: i⁵ = i, i⁶ = -1, i⁷ = -i, i⁸ = 1, and so on. This creates a repeating cycle of four values: i, -1, -i, 1.

Remember This: The powers of i follow a simple pattern: i, -1, -i, 1. If you need to find a higher power, divide the exponent by 4 and look at the remainder!

of 3

Multiplying Complex Numbers

Multiplying complex numbers requires applying the pattern of i powers. When you multiply expressions with i, remember that i² equals -1.

Let's try an example: (3) $2-7i$ + (4i) $2-7i$ . First, multiply 3 by each term in $2-7i$ to get 6-21i. Then multiply 4i by each term in $2-7i$ to get 8i-28i².

Since i² equals -1, we can rewrite 28i² as 28 $-1$ or -28. Combining all terms: 6-21i+8i+28 = 34-13i.

Pro Tip: When multiplying complex numbers, distribute first just like with regular algebraic expressions, then substitute i² = -1, i³ = -i, and i⁴ = 1 wherever needed!

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You can download the app in the Google Play Store and in the Apple App Store.

That's right! Enjoy free access to study content, connect with fellow students, and get instant help – all at your fingertips.

Students love us — and so will you.

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Understanding Complex Numbers

The Powers of i

Multiplying Complex Numbers

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