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Chapter 2: Problem 67

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 \(2-3i\) and multiplying \(2+3i\) by its complex conjugate yields 13.

Step by step solution

01

Finding the Complex Conjugate

A complex number has the form \(a+bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The complex conjugate of a complex number is the number with the same real part and negated imaginary part. In this case, the complex conjugate of \(2+3i\) is \(2-3i\).
02

Multiplying the Complex Number and its Conjugate

Next, the multiplication of the complex number and its conjugate has to be executed. This results in \((2+3i)*(2-3i)\). In order to get the result, one has to use the distributive property twice. First, multiply \(2*(2-3i)\) to get \(4-6i\). Then, multiply \(3i*(2-3i)\) to get \(6i-9i^2\). However, remember that \(i^2\) is -1, so it becomes \(6i+9\). Adding the two results together gives \(4-6i+6i+9\) resulting in \(13+0i\) or just \(13\).
03

Conclusion

Multiplying a complex number by its complex conjugate results in a real number. This is always the case due to the property of imaginary unit \(i\). In this case, the conjugate of \(2+3i\) is \(2-3i\), and the result of their multiplication is 13.

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