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 \(z z ^ { * } - | z | ^ { 2 } = 0\).

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

1 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.
[0pt] [1 mark] $$\left( z ^ { * } \right) ^ { * } = z \quad z z ^ { * } = | z | ^ { 2 } \quad ( - z ) ^ { * } = - \left( z ^ { * } \right) \quad z - z ^ { * } = z ^ { * } - z$$