Problem 76 - Contemporary Precalculus

Question

Let \(z=a+b i\) be a complex number and denote its conjugate \(a-b i\) by \(\bar{z} .\) Prove that \(|z|^{2}=z \bar{z}\).

Step-by-Step Solution

Answer

Question: Prove that for a complex number z and its conjugate, the square of the magnitude of z is equal to the product of z and its conjugate. Answer: Using the steps provided in the solution, we proved that for a complex number \(z = a + bi\) and its conjugate \(\bar{z} = a - bi\), the following equality holds: \(|z|^2 = z\bar{z}\).

Step 1: Rewrite the Complex Number and its Conjugate

Given a complex number \(z = a + bi\), its conjugate is denoted as \(\bar{z} = a - bi\), where \(a\) and \(b\) are real numbers and \(i^2 = -1\).

Step 2: Recall the Magnitude of a Complex Number

The magnitude (or absolute value) of the complex number \(z = a + bi\) is given by \(|z| = \sqrt{a^2 + b^2}\).

Step 3: Multiply the Complex Number and its Conjugate

The product of \(z\) and its conjugate \(\bar{z}\) is given by: \(z\bar{z} = (a + bi)(a - bi)\)

Step 4: Simplify the Product

Using the distributive property (or FOIL method), we get: \(z\bar{z} = a^2 - abi + abi - b^2i^2\) Notice that the two middle terms, \(-abi\) and \(+abi\), cancel each other out. Additionally, since \(i^2 = -1\), we can simplify the expression further to: \(z\bar{z} = a^2 - b^2(-1)\)

Step 5: Prove the Equality

Finally, we simplify the expression to: \(z\bar{z} = a^2 + b^2\) From Step 2, we know that the square of the magnitude of the complex number \(z\) is: \(|z|^2 = (a^2 + b^2)\) Thus, we have proven that for a complex number \(z = a + bi\) and its conjugate \(\bar{z} = a - bi\), the following equality holds: \(|z|^2 = z\bar{z}\)

Key Concepts

Magnitude of Complex Numbers

The magnitude of a complex number is sometimes called its absolute value. It measures how "long" the complex number is if you imagine it as a vector on the complex plane. For a complex number given by \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with \( i^2 = -1 \), its magnitude is calculated using the formula:

• \(|z| = \sqrt{a^2 + b^2}\)

This formula might remind you of the Pythagorean theorem from geometry. Imagine \( a \) and \( b \) as the legs of a right triangle, with \( |z| \) being the hypotenuse.

This connection to right triangles helps to understand why the magnitude is calculated in this way. The complex number's position on the complex plane and its distance from the origin can be quickly assessed by evaluating \(|z|\). It's a useful tool for measuring the size or amplitude of a complex number.

Complex Conjugates

Complex conjugates are closely related to the idea of complex numbers. If you have a complex number \( z = a + bi \), its conjugate, denoted by \( \bar{z} \), is \( a - bi \). The only difference is the sign of the imaginary part. This small change can have significant impacts in calculations.

Why are complex conjugates important? Here's why:

• They simplify the process of dividing complex numbers.
• They facilitate finding the magnitude of a complex number squared, as in the given proof.
• They help in simplifying expressions, as the imaginary parts cancel each other out when a complex number is multiplied by its conjugate.

Complex conjugates have the property of making calculations with complex numbers more straightforward, especially when concerning products and divisions. These concepts show elegant symmetry inherent in complex numbers, which makes many operations more manageable.

Proofs in Mathematics

Mathematical proofs are essential for establishing truths within the discipline. They provide a structured, logical argument to demonstrate why a particular statement is universally valid. In the context of complex numbers, proofs often appear when we establish identities or validate properties, like the one in the exercise: \(|z|^2 = z\bar{z}\).Proofs make use of properties and definitions that are already known to be true. In this exercise, properties of algebra and the definition of a complex number's magnitude were employed to show the equality of \(|z|^2\) and \(z\bar{z}\).

Here are reasons proofs are particularly important:

• They assure that mathematical statements are no mere conjectures but concretely verified truths.
• Proofs help us understand why certain mathematical properties hold, thus deepening comprehension.
• They provide rigor and reliability to mathematical exploration and innovation.

Learning how to follow and create proofs is a vital skill in mathematics. It allows a deeper engagement with mathematical concepts, ensuring that the foundations of knowledge are strong and well-understood.