Complex conjugate properties and proofs

3

1. Given the complex numbers \(u = a + \mathrm { i } b\) and \(w = c + \mathrm { i } d\), where \(a , b , c\) and \(d\) are real, prove that \(( u + w ) ^ { * } = u ^ { * } + w ^ { * }\).
2. Solve the equation \(( z + 2 + \mathrm { i } ) ^ { * } + ( 2 + \mathrm { i } ) z = 0\), giving your answer in the form \(x + \mathrm { i } y\) where \(x\) and \(y\) are real.

2

1. Given that \(z = a + b \mathrm { j }\), express \(| z |\) and \(z ^ { * }\) in terms of \(a\) and \(b\).
2. Prove that \(z z ^ { * } - | z | ^ { 2 } = 0\).

7 Given that \(z\) is a complex number, prove that \(z z ^ { * } = | z | ^ { 2 }\).

\(z = 3 - i\) Determine the value of \(zz*\) Circle your answer. [1 mark] \(10\) \(\qquad\) \(\sqrt{10}\) \(\qquad\) \(10 - 2i\) \(\qquad\) \(10 + 2i\)

Given that \(z\) is a complex number, and that \(z^*\) is the complex conjugate of \(z\), which of the following statements is not always true? Circle your answer. [1 mark] \((z^*)^* = z\) \quad\quad \(zz^* = |z|^2\) \quad\quad \((-z)^* = -(z^*)\) \quad\quad \(z - z^* = z^* - z\)

Given that \(z\) is a complex number and that \(z^*\) is the complex conjugate of \(z\) prove that \(zz^* - |z|^2 = 0\) [3 marks]