Problem 61 - Algebra and Trigonometry

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 when they are multiplied, the result is 13.

Step 1: Find the Complex Conjugate

The complex conjugate of a complex number has the same real part and the imaginary part with the opposite sign. So, the complex conjugate of \(2+3i\) is \(2-3i\)

Step 2: Multiply the Complex Number by its Conjugate

To multiply the original number with its conjugate, we use the algebraic rule \((a+b) * (a-b) = a^2 - b^2\) resulting in \( (2+3i) * (2-3i) = 2^2 - (3i)^2 = 4 - (-9) = 4 + 9 = 13 \)

Step 3: Answer

Therefore, when \(2+3i\) is multiplied by its complex conjugate \(2-3i\), we get 13