What is the product of a + bi and a - bi where a and b are real numbers, a ≠ 0, and b ≠ 0? Classify the product as a real or an imaginary number. Explain.

What is the product of a + bi and a - bi where a and b are real numbers, a ≠ 0, and b ≠ 0? Classify the product as a real or an imaginary number. Explain.

Question
What is the product of a + bi and a - bi where a and b are real numbers, \(\displaystyle{a}≠{0}\), and \(\displaystyle{b}≠{0}\)? Classify the product as a real or an imaginary number. Explain.

Answers (1)

2020-10-26
One way to find the product is by using the rule: \(\displaystyle{\left({x}+{y}\right)}{\left({x}−{y}\right)}={x}^{{2}}−{y}^{{2}}\)
\(\displaystyle{\left({a}+{b}{i}\right)}{\left({a}−{b}{i}\right)}={a}^{{2}}+{\left({b}{i}\right)}^{{2}}={a}^{{2}}+{\left(−{b}^{{2}}\right)}={a}^{{2}}−{b}^{{2}}\) Since aa and b are both real numbers, then \(\displaystyle{a}^{{2}}−{b}^{{2}}\) is a real number.
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