What Are Imaginary Numbers? Calculating, Rules & Examples

Lesson Overview

• What Are Imaginary Numbers?
• Imaginary Number Rules
• Calculating Imaginary Numbers
• Adding and Subtracting Imaginary Numbers
• Multiplying Imaginary Numbers
• Simplifying Powers of i
• Dividing Imaginary Numbers
• Examples on Imaginary Numbers

Imaginary numbers are numbers that, when squared, result in negative values. They extend the number system to solve equations that real numbers cannot and are crucial in various fields such as engineering and physics.

What Are Imaginary Numbers?

Imaginary numbers are numbers that cannot be represented on the real number line. These numbers are based on the unit "i," where i is the square root of the negative one (i² = -1). 

Example: Square root of -9

1. Start with the expression: √(-9)
2. Rewrite it as: √(9) × √(-1)
3. Simplify √(9) = 3, and √(-1) = i
4. Combine the results: 3i

Thus, the square root of -9 is 3i. This shows how imaginary numbers allow us to work with the square roots of negative numbers.

Imaginary Number Rules

Key Rules for Imaginary Numbers

• i = √-1
• i² = -1
• i³ = -i
• i⁴ = 1
• i⁴ⁿ = 1 (for any integer n)
• i⁴ⁿ⁻¹ = -i (for any integer n)

Remember these points:

1. Imaginary numbers have specific rules to simplify calculations. A complex number is expressed as a + bi, with "a" as the real part and "bi" as the imaginary part.
2. The conjugate of a + bi is a - bi. Complex roots always come in conjugate pairs, and multiplying them gives equations with real coefficients.
3. For example, solving x² = a (where a is a real number) may involve taking the square root of a, which can result in imaginary solutions.

The key rules for imaginary numbers are as follows:

• i = √-1
• i² = -1
• i³ = -i
• i⁴ = 1
• i⁴ⁿ = 1 (for any integer n)
• i⁴ⁿ⁻¹ = -i (for any integer n)

Calculating Imaginary Numbers

Imaginary numbers are calculated using rules similar to real numbers, with specific guidelines for handling i. Below are the methods for basic operations with imaginary numbers:

Adding and Subtracting Imaginary Numbers

Combine like terms as in algebra.

Example: 7i - 4i = 3i

Example: 3i + 2i = 5i

Multiplying Imaginary Numbers

Use standard multiplication rules and simplify using i squared equals negative one.

Example: 2i squared times -3i cubed = -6i to the power of 5 = -6 times i to the power of 4 times i = -6 times 1 times i = -6i

Example: 4i times 3i = 12i squared = 12 times -1 = -12

Simplifying Powers of i

The powers of i follow a repeating pattern:

• i to the power of 4k = 1
• i to the power of 4k plus 1 = i
• i to the power of 4k plus 2 = -1
• i to the power of 4k plus 3 = -i, where k is a whole number

Examples:

• i to the power of 8 = i to the power of 4 times 2 = 1
• i to the power of 13 = i to the power of 4 times 3 plus 1 = i

Dividing Imaginary Numbers

Apply the rule a to the power of m divided by a to the power of n equals a to the power of m minus n. If i is in the denominator, rationalize it using 1 divided by i equals -i.

• Example: 8i divided by 2i = 4
• Example: 6i cubed divided by 3i squared = 2i

Examples on Imaginary Numbers

Example 1: Simplifying powers of i

Simplify i¹⁵

• Solution:
  • Divide 15 by 4: 15 ÷ 4 = 3 remainder 3
  • This means i¹⁵ = i⁴⁺³
  • Using the rule i⁴ⁿ⁺³ = -i, we get i¹⁵ = -i

Example 2: Multiplying imaginary numbers

Simplify (2i)(3i)

• Solution:
  • (2i)(3i) = 6i²
  • 6i² = 6(-1) = -6

Example 3: Adding complex numbers

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

• Solution:
  • (3 + 2i) + (5 - 4i) = (3 + 5) + (2 - 4)i
  • = 8 - 2i

Example 4: Subtracting complex numbers

Simplify (7 - i) - (3 + 5i)

• Solution:
  • (7 - i) - (3 + 5i) = (7 - 3) + (-1 - 5)i
  • = 4 - 6i

Example 5: Multiplying complex numbers

Simplify (2 + i)(3 - 2i)

• Solution:
  • Use the distributive property (FOIL): (2 + i)(3 - 2i) = 6 - 4i + 3i - 2i²
  • Simplify: 6 - 4i + 3i - 2(-1)
  • Combine terms: 6 - i + 2 = 8 - i