Question

Prove that \(z \bar{z}=|z|^{2}\). Deduce that \(z^{-1}=\bar{z} /|z|^{2}\) if \(z \neq 0 .{ }^{4}\)

Short answer

The formula \(z \bar{z} = |z|^2\) holds for any complex \(z\); if \(z \neq 0\), then \(z^{-1} = \bar{z} / |z|^2\).

Step-by-step solution

Understanding the Problem

We need to prove that for any complex number, the product of the number and its complex conjugate equals the square of its modulus. Additionally, we need to show if \(z\) is not zero, then the inverse of \(z\) is equal to the complex conjugate of \(z\) divided by the square of its modulus.

Defining the Complex Number

Let \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The complex conjugate of \(z\) is \(\bar{z} = a - bi\).

Calculating the Product of the Complex Number and its Conjugate

Compute \(z \bar{z}\). We have: \(z \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\).

Finding the Modulus and its Square

The modulus of \(z\) is \(|z| = \sqrt{a^2 + b^2}\). Thus, \(|z|^2 = (\sqrt{a^2 + b^2})^2 = a^2 + b^2\).

Verifying the Product Equals the Square of the Modulus

From the calculations, \(z \bar{z} = a^2 + b^2\) and \(|z|^2 = a^2 + b^2\). Therefore, \(z \bar{z} = |z|^2\). This proves the first part of the exercise.

Expressing the Inverse of a Non-Zero Complex Number

If \(z eq 0\), the inverse \(z^{-1}\) is given by dividing \(1\) by \(z\): \(z^{-1} = \frac{1}{z}\). We need to express this in terms of \(\bar{z}\) and \(|z|^2\).

Using the Conjugate to Find the Inverse

Multiply both numerator and denominator of \(\frac{1}{z}\) by \(\bar{z}\): \(z^{-1} = \frac{\bar{z}}{z \bar{z}} = \frac{\bar{z}}{|z|^2}\). Hence, \(z^{-1} = \bar{z} / |z|^2\) when \(z eq 0\).

Key concepts

Complex Conjugate

Complex numbers are numbers that have both a real and an imaginary part, typically written as \(z = a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The complex conjugate of a complex number is essentially a reflection of the number over the real axis. For any complex number \(z\), its conjugate is denoted as \(\bar{z}\) and defined as \(\bar{z} = a - bi\). This means we simply change the sign of the imaginary part while keeping the real part the same.

Why is the complex conjugate important? When multiplying a complex number by its conjugate, the imaginary parts cancel out, resulting in a real number. This property is crucial when dealing with division of complex numbers or finding the inverse. For example, multiplying \(z\) and \(\bar{z}\) gives \((a + bi)(a - bi) = a^2 + b^2\), which is a real number.

Modulus of a Complex Number

The modulus of a complex number \(z = a + bi\) is a measure of its size or magnitude. It's computed as the distance from the origin to the point \((a, b)\) in the complex plane. Mathematically, this is defined by \(|z| = \sqrt{a^2 + b^2}\).

• The modulus gives us a single non-negative real number, representing how large the complex number is.
• It is always non-negative, meaning it is either positive or zero.

The square of this modulus, \(|z|^2\), simplifies the calculation especially when dealing with products of complex numbers as shown in the exercise: \(z \bar{z} = a^2 + b^2 = |z|^2\). This equivalence is central to proving properties related to complex numbers.

Inverse of a Complex Number

Finding the inverse of a complex number is akin to finding a reciprocal in real numbers but requires a different approach due to the presence of the imaginary unit \(i\). For a non-zero complex number \(z = a + bi\), its inverse is given by \(z^{-1} = \frac{1}{z}\), but we express it more elegantly as \(z^{-1} = \frac{\bar{z}}{|z|^2}\).

This formulation stems from multiplying both the numerator and denominator by the conjugate \(\bar{z}\), ensuring the denominator becomes a real number, specifically the square of the modulus \(|z|^2 = a^2 + b^2\).

Understanding this form is crucial, for:

• It simplifies division operations involving complex numbers.
• It uses the properties of conjugates and moduli to maintain real-valued denominators.

Thus, this inverse formula leverages basic algebraic properties of complex numbers to make operations seamless and efficient.