(7-5i)(7+5i)

2 min read Jun 16, 2024

Multiplying Complex Numbers: (7 - 5i)(7 + 5i)

This article will demonstrate how to multiply the complex numbers (7 - 5i) and (7 + 5i).

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, defined as the square root of -1 (i.e., i² = -1).

Multiplication of Complex Numbers

To multiply complex numbers, we use the distributive property (also known as FOIL - First, Outer, Inner, Last).

Multiplying (7 - 5i)(7 + 5i)

Let's multiply the complex numbers step-by-step:

First: 7 * 7 = 49
Outer: 7 * 5i = 35i
Inner: -5i * 7 = -35i
Last: -5i * 5i = -25i²

Now, let's combine the terms:

49 + 35i - 35i - 25i²

Since i² = -1, we can substitute:

49 + 35i - 35i - 25(-1)

Simplifying the expression:

49 + 25 = 74

Result

Therefore, the product of (7 - 5i) and (7 + 5i) is 74. This is a real number, which is interesting to note as the original numbers were complex. This occurs because the two complex numbers are complex conjugates - they have the same real part and opposite imaginary parts. Multiplying complex conjugates always results in a real number.