Problem 48 - College Algebra

Question

What is the complex conjugate of \(2+3 i ?\) What happens when you multiply this complex number by its complex conjugate?

Step-by-Step Solution

Answer

The complex conjugate of \(2+3i\) is \(2-3i\), and multiplying this complex number by its conjugate results in -5.

Step 1: Finding the Complex Conjugate

A complex number can be represented in the form \(a+bi\), where \(a\) and \(b\) are real numbers and \(i\) is the square root of -1. The complex conjugate of \(a+bi\) is \(a-bi\). Thus, to find the complex conjugate of \(2+3i\), we simply need to change the sign of the imaginary part.

Step 2: Calculation

Changing the sign of the imaginary part in \(2+3i\) leads to \(2-3i\). This is the complex conjugate.

Step 3: Multiplying the Complex Number and its Conjugate

To multiply the complex number \(2+3i\) and its conjugate \(2-3i\), we simply apply the rule of 'FOIL' as with binomials. Thus, we calculate \( (2+3i)*(2-3i) = (2*2) + (2*-3i) +(3i*2)+ (3i*-3i)\).

Step 4: Simplifying the Result

Simplifying the expression from step 3 results in \(4 - 6i + 6i -9 = 4 -9 = -5\) because \(i^2 = -1\).