Problem 68 - Algebra and Trigonometry

Question

Recall that the symbol \(\overline{z}\) represents the complex con- jugate of \(z .\) If \(z=a+b i\) and \(w=c+d i,\) prove each statement. $$ \overline{z w}=\overline{z} \cdot \overline{w} $$

Step-by-Step Solution

Answer

Thus, \(\overline{z w} = \overline{z} \cdot \overline{w}\) holds true.

Step 1: Write out the expressions for zw and conjugates

Given two complex numbers \(z = a + bi\) and \(w = c + di\), the product \(zw\) is computed as follows: \(zw = (a + bi)(c + di) = ac + adi + bci + bdi^2\). Since \(i^2 = -1\), this simplifies to \(zw = (ac - bd) + (ad + bc)i\). Next, we need the conjugate of \(zw\), which is \(\overline{zw} = (ac - bd) - (ad + bc)i\).

Step 2: Find the conjugates of z and w individually

Start with \(z = a + bi\), its conjugate is \(\overline{z} = a - bi\). For \(w = c + di\), its conjugate is \(\overline{w} = c - di\).

Step 3: Compute the product of the conjugates

Now calculate the product \(\overline{z} \cdot \overline{w} = (a - bi)(c - di)\). Use the distributive property to expand: \((a - bi)(c - di) = ac - adi - bci + bdi^2\). Simplify using \(i^2 = -1\) to get \(ac - bd - (ad + bc)i\).

Step 4: Compare the expressions

Both expressions \(\overline{zw} = (ac - bd) - (ad + bc)i\) and \(\overline{z} \cdot \overline{w} = (ac - bd) - (ad + bc)i\) are identical. Thus, \(\overline{z w} = \overline{z} \cdot \overline{w}\).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers and are typically denoted by the letter \(z\). They have the form \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The part \(a\) is known as the real part, and \(bi\) as the imaginary part.

• The real part aids in positioning the number along the horizontal axis of a complex plane.
• The imaginary part determines the number's position along the vertical axis.

A complex number can have infinite representations based on the values of \(a\) and \(b\). This unique representation allows for their use in various mathematical domains, such as engineering and physics. Understanding the nature of complex numbers enhances your ability to solve quadratic equations that lack real solutions.

Conjugate Multiplication

Conjugate multiplication involves multiplying the conjugates of two complex numbers. For a given complex number \(z = a + bi\), its conjugate is \(\overline{z} = a - bi\). When dealing with two complex numbers, \(z = a + bi\) and \(w = c + di\), their conjugate multiplication is expressed as \(\overline{z} \cdot \overline{w}\).

• First, calculate \(z \cdot w\) and take the conjugate of the result: \(\overline{zw}\).
• Then compute \(\overline{z} \cdot \overline{w}\) directly.

The exercise shows that both \(\overline{zw}\) and \(\overline{z} \cdot \overline{w}\) yield the same expression. This equality states that the conjugate of a product is the product of the conjugates. It's a property that solidifies our understanding of how complex numbers interact under multiplication.

Imaginary Unit Properties

The imaginary unit, denoted by \(i\), is defined by the property \(i^2 = -1\). This property is crucial when working with complex numbers, especially in calculations involving conjugates and multiplication.

• In the multiplication of complex numbers, like \((a + bi)(c + di)\), the term \(bdi^2\) simplifies to \(-bd\).
• This transformation stems directly from the fundamental property of \(i\), facilitating the simplification of expressions involving \(i\).

In conjugate multiplication, setting \(i^2 = -1\) allows us to resolve complex terms into a simplified form that highlights their real and imaginary components distinctly. Mastering this property aids in diverse mathematical operations that utilize complex numbers.