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Chapter 2: Problem 45

Write the complex conjugate of the complex number. Then multiply the number by its complex conjugate. $$-1-\sqrt{5} i$$

Short Answer

Expert verified
The complex conjugate of the number \(-1-\sqrt{5}i\) is \(-1+\sqrt{5}i\). After multiplying the two, the result is \(-4\).

Step by step solution

01

Finding the Complex Conjugate

The complex conjugate changes the sign of the imaginary part of the complex number. Given that the number is \(-1-\sqrt{5}i\), its complex conjugate will be \(-1+\sqrt{5}i\).
02

Multiplying by the Complex Conjugate

To multiply two complex numbers, use the FOIL method 鈥� multiply the first elements, multiply the outer elements, multiply the inner elements, and multiply the last elements from each parentheses. After doing this with \(-1-\sqrt{5}i\) and \(-1+\sqrt{5}i\), we get \(1-\sqrt{5}i+\sqrt{5}i-5\). Notice that the two middle terms cancel out.
03

Simplifying the Expression

The result of the multiplication simplifies down to \(1-5\), which equals \(-4\).

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Most popular questions from this chapter

Prove that the complex conjugate of the product of two complex numbers \(a_{1}+b_{1} i\) and \(a_{2}+b_{2} i\) is the product of their complex conjugates.

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