(2+5i)(2-5i)

3 min read Jun 16, 2024

Understanding Complex Number Multiplication: (2 + 5i)(2 - 5i)

This article will explore the multiplication of complex numbers, specifically focusing on the expression (2 + 5i)(2 - 5i).

What are Complex Numbers?

Complex numbers extend the real number system by incorporating the imaginary unit i, where i² = -1. They are expressed in the form a + bi, where a and b are real numbers.

Multiplying Complex Numbers

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

Let's break down the multiplication of (2 + 5i)(2 - 5i):

First: 2 * 2 = 4
Outer: 2 * (-5i) = -10i
Inner: 5i * 2 = 10i
Last: 5i * (-5i) = -25i²

Now, we combine the terms and remember that i² = -1:

4 - 10i + 10i - 25(-1) = 4 + 25 = 29

Therefore, (2 + 5i)(2 - 5i) = 29.

Significance of the Result

This particular multiplication demonstrates a crucial concept in complex numbers:

(a + bi)(a - bi) = a² + b²

This pattern reveals that the product of a complex number and its conjugate (the complex number with the opposite sign of the imaginary part) always results in a real number. In this case, the conjugate of (2 + 5i) is (2 - 5i).

Conclusion

Understanding complex number multiplication is essential for various mathematical applications, including solving equations, analyzing circuits, and working with wave phenomena. By employing the distributive property and recognizing the special case of multiplying a complex number by its conjugate, we can effectively perform these calculations.