Simplifying Using Complex Conjugate

00:00 4 marks

Let \(z = 3 + 4i\) be a complex number.

(a) Define the complex conjugate of \(z\).

(b) Use your definition to simplify the expression \(\frac{1}{3+4i}\) by multiplying numerator and denominator by the complex conjugate of \(z\).

Part a

2 minutes 1 marks

Question

Define the complex conjugate of a complex number \(z = a+bi\).

Hint Hint

Hint

Recall that the conjugate of \(a+bi\) is \(a-bi\).

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

The complex conjugate of \(z = a+bi\) is \(\overline{z} = a-bi\).

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet

Part b

3 minutes 3 marks

Question

Simplify the expression \(\frac{1}{3+4i}\) by multiplying numerator and denominator by the complex conjugate of \(3+4i\). Express your answer in the form \(a+bi\).

Hint Hint

Hint

Multiply numerator and denominator by \(3-4i\).

Answer Answer
Mark Scheme Mark Scheme
Explanation Explanation
Formula Booklet

Answer Answer

Complete Answer

\(\frac{1}{3+4i} = \frac{3-4i}{25} = \frac{3}{25} – \frac{4}{25}i\)

Mark Scheme Mark Scheme

Answer Explanation Answer Explanation

Formula Booklet