Problem 115 - Calculus for Biology and Medicine

Question

Show \(\overline{z w}=\bar{z} \bar{w}\).

Step-by-Step Solution

Answer

The conjugate of a product equals the product of the conjugates: \(\overline{z w} = \overline{z} \overline{w}\).

Step 1: Understand Complex Conjugates

To solve this, first recall the definition of the complex conjugate. If we have a complex number \(z = a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit, the complex conjugate \(\overline{z}\) is \(a - bi\). Similarly, for another complex number \(w = c + di\), its conjugate \(\overline{w} = c - di\).

Step 2: Multiply the Complex Numbers

Next, multiply the complex numbers \(z\) and \(w\). If \(z = a + bi\) and \(w = c + di\), then \(zw = (a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i\), using \(i^2 = -1\).

Step 3: Conjugate the Product

Take the conjugate of the product \(zw = (ac - bd) + (ad + bc)i\). The conjugate is \(\overline{zw} = (ac - bd) - (ad + bc)i\).

Step 4: Multiply the Conjugates Individually

Now, multiply the conjugates \(\overline{z}\) and \(\overline{w}\). We have \(\overline{z} = a - bi\) and \(\overline{w} = c - di\), so \(\overline{z} \cdot \overline{w} = (a - bi)(c - di) = ac - adi - bci + bdi^2 = (ac - bd) - (ad + bc)i\). Again, use \(i^2 = -1\).

Step 5: Compare the Results

Notice that in Step 3, the conjugate of the product is \((ac - bd) - (ad + bc)i\). In Step 4, the product of the conjugates is also \((ac - bd) - (ad + bc)i\). Therefore, the two expressions are equal: \(\overline{zw} = \overline{z} \cdot \overline{w}\).

Key Concepts

Understanding Complex Numbers

Complex numbers are like a two-in-one combo, featuring both a real part and an imaginary part. Imagine them as ordered pairs composed of a real number and an imaginary number formulated with the imaginary unit, denoted by \( i \), where \( i^2 = -1 \). The basic format is \( z = a + bi \), where:

• \( a \) is the real part
• \( b \) is the imaginary part. It is crucial to remember that the imaginary component involves the unit \( i \)

When you visualize complex numbers on a plane, the real part represents the horizontal axis, and the imaginary part represents the vertical axis.
This forms a complex number into a point or a vector-like form in the coordinate grid, adding a useful level of abstraction and power to the number system suitable for various mathematical problems.

The Properties of Complex Conjugation

Complex conjugation is a fascinating property that attaches a reflective nature to complex numbers. If you imagine a complex number \( z = a + bi \), its complex conjugate is \( \overline{z} = a - bi \).
To better understand, imagine reflecting the complex number across the real axis of the complex plane.
The imaginary part flips its sign, while the real part stays untouched.

• This operation is valuable in simplifying complex expressions
• It assists in solving equations where complex numbers are involved

An interesting property of complex conjugation is that when you find the modulus squared of \( z \), it looks like this: \( |z|^2 = z \cdot \overline{z} = (a + bi)(a - bi) = a^2 + b^2 \).
No imaginary component due to how conjugation neutralizes the imaginary unit. Hence a helpful tool in various calculations.

Multiplication of Complex Numbers and Conjugation

Multiplication of complex numbers can initially seem a bit challenging, but it's a logical extension of the distributive property.
For two complex numbers, \( z = a + bi \) and \( w = c + di \), the product \( zw \) is obtained as follows: \[ zw = (a + bi)(c + di) = ac + adi + bci + bdi^2 \]
Remember, since \( i^2 = -1 \), the expression transforms to:
\( zw = (ac - bd) + (ad + bc)i \).
A captivating aspect is how multiplication and conjugation relate.

• Taking the conjugate of a product is the same as multiplying the conjugates individually
• This is expressed by \( \overline{zw} = \overline{z} \cdot \overline{w} \)

Once you understand this equality, it becomes a powerful identity useful in many areas of mathematics, particularly when simplifying expressions or solving equations that involve complex numbers.