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

2 min read Jun 16, 2024
(3+2i)(3-2i)

Multiplying Complex Numbers: (3 + 2i)(3 - 2i)

This article explores the multiplication of complex numbers, specifically focusing on the product of (3 + 2i) and (3 - 2i).

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.

Multiplication Process

To multiply complex numbers, we can use the distributive property (also known as FOIL method):

(3 + 2i)(3 - 2i) = 3(3) + 3(-2i) + 2i(3) + 2i(-2i)

Simplifying the expression:

= 9 - 6i + 6i - 4i²

Since i² = -1, we can substitute it in the equation:

= 9 - 4(-1)

= 9 + 4

= 13

Result and Significance

The product of (3 + 2i) and (3 - 2i) is 13, a real number. This demonstrates an important property of complex numbers:

  • The product of a complex number and its conjugate is always a real number.

The conjugate of a complex number a + bi is a - bi. In our example, (3 - 2i) is the conjugate of (3 + 2i).

This property is widely used in simplifying complex expressions and solving equations involving complex numbers.

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