Question

Austin was having trouble remembering how to find the conjugate of a complex number. He thought the conjugate of (7 – 4i) was (-7 + 4i). Tracey reminded him that if he multiplied a complex number by it’s conjugate that the term with the i in it would be eliminated. 1. Multiply (7 – 4i) • (-7 + 4i) to prove that they are not conjugates. Explain how your answer proves they are NOT conjugates. 2. What is the conjugate of (7 – 4i)? Show the multiplication to verify your answer.

Answer 1

Explanation:

for a conjugate of an expression you change the sign of one term. not both.

2.

the conjugate of (7 - 4i) that makes the i-term disappear is (7 + 4i).

(7 - 4i)(7 + 4i) = 7×7 + 7×4i - 7×4i - 4²i² = 49 + 16 = 65

1.

(7 - 4i)(-7 + 4i) = -7×7 + 7×4i + 7×4i - 4²i² = -49 + 56i + 16 =

= -33 + 56i

as we can see, there is still an i-term.

real conjugates of terms a and b create a multiplication product (a + b)(a - b) = (a² - b²).

in our case (7² - 4²i²) = (49 + 16) = 65.

that is not the same as -33 + 56i.