(−5 + 3i) ⋅ (1 − 2i)

Jasonbradley

2 min read

·

Jun 17, 2024

20

Share

Multiplying Complex Numbers: A Step-by-Step Guide

This article will guide you through the process of multiplying two complex numbers: (-5 + 3i) ⋅ (1 - 2i).

Understanding Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. The imaginary part is denoted by the letter i, where i² = -1. We represent complex numbers in the form a + bi, where 'a' is the real part and 'b' is the imaginary part.

The Multiplication Process

To multiply complex numbers, we use the distributive property, similar to multiplying binomials.

1. Expand the product: (-5 + 3i) ⋅ (1 - 2i) = (-5)(1) + (-5)(-2i) + (3i)(1) + (3i)(-2i)

2. Simplify each term: = -5 + 10i + 3i - 6i²

3. Substitute i² with -1: = -5 + 10i + 3i - 6(-1)

4. Combine real and imaginary terms: = (-5 + 6) + (10 + 3)i

5. Final result: = 1 + 13i

Therefore, the product of (-5 + 3i) ⋅ (1 - 2i) is 1 + 13i.

Key Takeaways

• When multiplying complex numbers, treat i as a variable and apply the distributive property.
• Remember that i² = -1.
• Combine real and imaginary terms to express the final result in the standard form a + bi.

This process helps you understand the basic operations involving complex numbers and lays the foundation for further exploration in complex number algebra.