Problem 3 - Complex Analysis for Mathematics and Engineering

Question

Show that \(z \bar{z}\) is always a real number.

Step-by-Step Solution

Answer

\( z \bar{z} = a^2 + b^2 \) is a real number.

Step 1: Define a Complex Number

A complex number \( z \) can be expressed in the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property \( i^2 = -1 \).

Step 2: Identify the Conjugate

The conjugate of a complex number \( z = a + bi \) is \( \bar{z} = a - bi \). It changes the sign of the imaginary part of the complex number.

Step 3: Multiply the Complex Number and Its Conjugate

Compute \( z \bar{z} \) using the previously defined \( z \) and \( \bar{z} \): \[ z \bar{z} = (a + bi)(a - bi). \]

Step 4: Use Algebraic Expansion

Expand the expression \((a + bi)(a - bi)\) using the formula \((x + y)(x - y) = x^2 - y^2\):\[ z \bar{z} = a^2 - (bi)^2. \]

Step 5: Simplify the Expression

Since \( (bi)^2 = b^2i^2 \), and \( i^2 = -1 \), we have \( (bi)^2 = -b^2 \). Substitute back into the expression: \[ z \bar{z} = a^2 + b^2. \]

Step 6: Conclude with Real Number Property

The expression \( a^2 + b^2 \) is the sum of squares of real numbers \( a \) and \( b \), which is always a non-negative real number.

Key Concepts

Conjugate of a Complex Number

In the fascinating world of complex numbers, every complex number has a mirror version called its conjugate. For a complex number expressed as \( z = a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit, the conjugate \( \bar{z} \) is simply \( a - bi \). This operation involves flipping the sign of the imaginary part, leaving the real part unchanged.

• Complex number example: \( z = 3 + 4i \) with conjugate \( \bar{z} = 3 - 4i \)
• More broadly, for any complex number \( z = a + bi \), conjugate \( \bar{z} \) is \( a - bi \).

Finding the conjugate is especially useful in various mathematical operations. It helps to simplify expressions and rationalize denominators when dealing with complex fractions.

Multiplication of Complex Numbers

Multiplying complex numbers might seem tricky at first, but it's a lot like expanding expressions when we first learned about real numbers. When we multiply a complex number by its conjugate, we use a special algebraic formula:

• If \( z = a + bi \) and \( \bar{z} = a - bi \), then the product is \( z\bar{z} = (a + bi)(a - bi) \).

Expanding this using the difference of squares formula \((x+y)(x-y) = x^2 - y^2\), we have \( z\bar{z} = a^2 - (bi)^2 \). Continuing, since \((bi)^2 = b^2i^2\) and \(i^2 = -1\), we simplify \((bi)^2 = -b^2\), so the final expression becomes \( a^2 + b^2 \). This result is a real number, showcasing one fascinating property of multiplying a complex number by its conjugate.

Real Numbers

Real numbers are the numbers we use every day - from counting apples to measuring distances. They include all the numbers along the number line: integers, fractions, and even pi or square roots.

• Real numbers make up the set of all rational and irrational numbers.

In our complex numbers context, real numbers appear as the sum of squares \( a^2 + b^2 \), when you multiply a complex number with its conjugate.
This expression results in a real number because it's simply a sum of squares, which can never be negative. Whether we have \( a^2 \) or \( b^2 \), both are inherently non-negative, so their addition remains non-negative.
By observing the multiplication result of \( z\bar{z} \), we notice it confirms the magnitude or modulus of the complex number and is always a real number.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a mathematical concept that enables us to extend the real number system. Its defining property is that when it is squared, it equals -1: \( i^2 = -1 \). This simple property allows mathematicians to expand the number system to include solutions to equations that don't have real-number solutions, such as the square root of a negative number.

• Key property: \( i^2 = -1 \)
• Example: \( \sqrt{-1} = i \)

In complex numbers, the imaginary unit is used to represent the "imaginary" part. For example, in \( z = a + bi \), \( bi \) is the imaginary component.
When multiplying complex numbers, understanding \( i^2 = -1 \) is crucial for simplifying products like \( (bi)^2 \), since it turns into \( -b^2 \). This transformation is a cornerstone for proving that \( z\bar{z} \) results in a real number when complex numbers are involved.