Problem 10 - A Gentle Introduction to the Art of Mathematics

Question

The conjugate of a complex number is denoted with a superscript star, and is formed by negating the imaginary part. Thus if \(z=3+4 i\) then the conjugate of \(z\) is \(z^{*}=3-4 i\). Give an argument as to why the product of a complex number and its conjugate is a real quantity. (I.e. the imaginary part of \(z \cdot z^{*}\) is necessarily \(0,\) no matter what complex number is used for \(z\).)

Step-by-Step Solution

Answer

The product of a complex number and its conjugate is real because \( (a+bi)(a-bi) = a^2 + b^2 \), with no imaginary part.

Step 1 - Understand the Definition of Conjugate

The conjugate of a complex number is obtained by changing the sign of its imaginary part. For any complex number defined as \(z = a + bi\), its conjugate is \(z^* = a - bi\), where \(a\) and \(b\) are real numbers.

Step 2 - Write the Product of the Complex Number and Its Conjugate

Consider the complex number \(z = a + bi\) and its conjugate \(z^* = a - bi\). The product \(z \times z^*\) is: \((a + bi) \times (a - bi)\).

Step 3 - Expand the Product Using the Distributive Property

Apply the distributive property (FOIL method) to expand the expression:\((a + bi)(a - bi) = a(a - bi) + bi(a - bi)\). This simplifies to \(a^2 - abi + abi - b^2i^2\).

Step 4 - Simplify the Expanded Expression

Combine like terms in the expanded expression:\(a^2 - abi + abi - b^2i^2 = a^2 - b^2i^2\). Note that \(i^2 = -1\), so this further simplifies to \(a^2 - b^2(-1)\) or \(a^2 + b^2\).

Step 5 - Conclude the Real Part

The expression \(a^2 + b^2\) contains no imaginary component, as both \(a^2\) and \(b^2\) are real numbers. Thus, the product of a complex number and its conjugate is always a real number.

Key Concepts

complex numbers

Complex numbers are a fundamental concept in mathematics, particularly in fields such as engineering and physics. A complex number is composed of two parts:
- A real part
- An imaginary part
The standard form of a complex number is written as \(z = a + bi\), where:
- \(a\) is the real part
- \(bi\) is the imaginary part, with \(i\) representing the square root of -1
Complex numbers are useful because they allow us to solve equations that have no real solutions. They play a key role in understanding electrical circuits, signal processing, and quantum mechanics among other applications.

conjugate of complex number

The conjugate of a complex number is formed by changing the sign of its imaginary part. For a complex number \(z = a + bi\), its conjugate is denoted as \(z^* = a - bi\). So, if you start with a complex number and you negate the imaginary part (change the '+' to a '-'), you get the conjugate.
Why is the conjugate important? When you multiply a complex number by its conjugate, the result is always a real number. This means the imaginary parts cancel out. For instance, if \(z = 3 + 4i\), then \(z^* = 3 - 4i\). The product \(z \times z^* = (3 + 4i)(3 - 4i)\) expands to \(3^2 - (4i)^2\), which simplifies to \(9 + 16\) since \(i^2 = -1\). Thus, \(z \times z^* = 25\), a real number. This property is fundamental in many mathematical operations and proofs.

real and imaginary parts

To fully understand complex numbers and their conjugates, it's important to distinguish between the real and imaginary parts. For any complex number \(z = a + bi\):
- The real part is \(a\)
- The imaginary part is \(bi\) (where the coefficient \(b\) is real, and \(i\) is the imaginary unit)
When dealing with the product of a complex number and its conjugate, focus on these parts separately:
\begin{itemize}

The product of the real parts adds to the final result.

The product of the imaginary parts must be handled carefully since \(i^2 = -1\).

In the provided example, the product \((a + bi) \times (a - bi)\) simplifies because the middle terms involving imaginary units cancel each other out, leaving only real components \(a^2\) and \(b^2\). This shows that the product of a complex number and its conjugate is always real, demonstrating the symmetry and uniqueness of complex numbers.