Complex Number Multiplication: A Step-by-Step Guide

In the realm of mathematics, complex numbers extend the real number system by incorporating the imaginary unit, denoted by 'i', where i² = -1. This expansion opens up a world of possibilities, allowing us to solve equations that were previously unsolvable within the real number system. One fundamental operation in complex number arithmetic is multiplication, which we will explore in detail in this guide.

Understanding Complex Numbers

A complex number is typically represented in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. The real part of the complex number is 'a', and the imaginary part is 'b'.

Multiplying Complex Numbers

The multiplication of complex numbers follows the distributive property, similar to multiplying binomials in algebra. To multiply two complex numbers, (a + bi) and (c + di), we follow these steps:

1. Expand the product: (a + bi)(c + di) = ac + adi + bci + bdi²
2. Substitute i² with -1: ac + adi + bci + bdi² = ac + adi + bci - bd
3. Combine real and imaginary terms: (ac - bd) + (ad + bc)i

The result is a new complex number, where the real part is (ac - bd) and the imaginary part is (ad + bc).

Example

Let's multiply the complex numbers (2 + 3i) and (1 - 4i):

1. (2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i²
2. 2 - 8i + 3i - 12i² = 2 - 8i + 3i + 12
3. 2 - 8i + 3i + 12 = (2 + 12) + (-8 + 3)i = 14 - 5i

Therefore, (2 + 3i)(1 - 4i) = 14 - 5i.

Practice Problems

Here are a few practice problems to solidify your understanding of complex number multiplication:

1. (3 + 2i)(4 - i)
2. (5 - i)(2 + 3i)
3. (-1 + 6i)(3 + 2i)

Key Points

• Complex number multiplication involves expanding the product and substituting i² with -1.
• The result of multiplying two complex numbers is another complex number.
• Practice is key to mastering complex number multiplication.

Conclusion

Complex number multiplication is a fundamental operation in complex number arithmetic. Understanding this process is crucial for solving equations, exploring complex number properties, and applying complex numbers in various fields such as physics, engineering, and signal processing. By following the steps outlined in this guide and practicing diligently, you can gain a solid understanding of this essential mathematical concept.