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Chapter 1: Problem 61

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

Short Answer

Expert verified
The complex conjugate of \(2+3i\) is \(2-3i\). When this complex number is multiplied by its complex conjugate, the result is the real number 13.

Step by step solution

01

Find the Complex Conjugate

The given complex number is \(2+3i\). The complex conjugate is found by changing the sign of the imaginary part. So, the complex conjugate of \(2+3i\) is \(2-3i\).
02

Multiplication of Complex Number by its Conjugate

Now, we are asked to multiply the complex number \(2+3i\) by its complex conjugate \(2-3i\). You will do this as normal multiplication. Expand and simplify the result:\[(2 + 3i)(2 - 3i) = 4 - 6i + 6i - 9i^2.\]Remember that \(i^2 = -1\). So, replacing \(i^2\) with -1 gives: \[4 - 9(-1) = 4 + 9 = 13.\]

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