(1+3i)(1-3i)-2

Jasonbradley

2 min read

·

Jun 16, 2024

14

Share

Simplifying Complex Expressions: (1 + 3i)(1 - 3i) - 2

This article explores how to simplify the complex expression (1 + 3i)(1 - 3i) - 2. We'll break down the steps involved and understand the concepts behind them.

Understanding Complex Numbers

Complex numbers are numbers that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit. The imaginary unit i is defined as the square root of -1 (i.e., i² = -1).

Simplifying the Expression

1. Expand the product:

   • We use the distributive property (or FOIL method) to multiply the complex numbers: (1 + 3i)(1 - 3i) = 1(1) + 1(-3i) + 3i(1) + 3i(-3i) = 1 - 3i + 3i - 9i²

2. Substitute i² = -1:

   • Replace i² with -1 in the expression: 1 - 3i + 3i - 9i² = 1 - 3i + 3i - 9(-1)

3. Simplify:

   • Combine the real and imaginary terms: 1 - 3i + 3i - 9(-1) = 1 + 9 = 10

4. Subtract 2:

   • Finally, subtract 2 from the simplified result: 10 - 2 = 8

Conclusion

Therefore, the simplified form of the complex expression (1 + 3i)(1 - 3i) - 2 is 8. This example demonstrates the fundamental operations involved in working with complex numbers. Understanding the properties of complex numbers and applying basic algebraic techniques enables us to simplify expressions effectively.