Problem 74 - Algebra and Trigonometry

Question

Recall that the symbol \(\overline{z}\) represents the complex conjugate 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

The equality \( \overline{zw} = \overline{z} \cdot \overline{w} \) holds true for complex numbers.

Step 1: Define Complex Numbers

Let the complex numbers be defined as \( z = a + bi \) and \( w = c + di \), where \( a, b, c, \) and \( d \) are real numbers, and \( i \) is the imaginary unit such that \( i^2 = -1 \).

Step 2: Compute zw

Calculate the product \( zw = (a + bi)(c + di) \). Expanding this, \[ zw = ac + adi + bci + bdi^2 \]. Since \( i^2 = -1 \), this simplifies to \( zw = (ac - bd) + (ad + bc)i \).

Step 3: Compute Conjugate of zw

The conjugate of \( zw \), denoted as \( \overline{zw} \), replaces the imaginary part with its negative. Hence, \( \overline{zw} = (ac - bd) - (ad + bc)i \).

Step 4: Compute Conjugates of z and w

Find the conjugates of \( z \) and \( w \):\( \overline{z} = a - bi \) and \( \overline{w} = c - di \).

Step 5: Calculate the Product of Conjugates

Calculate the product \( \overline{z} \cdot \overline{w} = (a - bi)(c - di) \). Expanding, \[ \overline{z} \cdot \overline{w} = ac - adi - bci + bdi^2 \]. Since \( i^2 = -1 \), this simplifies to \( \overline{z} \cdot \overline{w} = (ac - bd) - (ad + bc)i \).

Step 6: Compare \( \overline{zw} \) and \( \overline{z} \cdot \overline{w} \)

Notice that \( \overline{zw} = (ac-bd) - (ad+bc)i \) is identical to \( \overline{z} \cdot \overline{w} = (ac-bd) - (ad+bc)i \). Thus, it is proven that \( \overline{zw} = \overline{z} \cdot \overline{w} \).

Key Concepts

Complex Numbers

Complex numbers are an extension of the real numbers and can be expressed in the form \(z = a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The number \(a\) is called the real part, and \(b\) is the imaginary part of the complex number. Complex numbers are often used in advanced mathematics and engineering fields.

• Real Part: The \(a\) in \(z = a + bi\)
• Imaginary Part: The \(b\) in \(z = a + bi\)
• Imaginary Unit: \(i\), where \(i^2 = -1\)

Understanding complex numbers allows us to perform arithmetic operations (addition, subtraction, multiplication, and division) and move easily between the real and imaginary components.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a mathematical concept that extends the real number system to include solutions to equations like \(x^2 = -1\). By definition, the property \(i^2 = -1\) holds for this unit. This concept is crucial for the construction of complex numbers and plays a significant role in various fields beyond mathematics, such as physics and electrical engineering.

Consider the following:

• The imaginary unit is not a real number itself, but it's a tool to facilitate computations involving square roots of negative numbers.
• Combined with the real number line, it forms the complex plane, which is a two-dimensional plane for graphing complex numbers.

Understanding the imaginary unit helps in grasping how complex numbers are structured and manipulated.

Product of Conjugates

The product of complex conjugates is a fundamental concept in complex analysis. If a complex number is given by \(z = a + bi\), then its complex conjugate is \(\overline{z} = a - bi\).

To explore the product of two complex conjugates, consider two complex numbers \(z = a + bi\) and \(w = c + di\). Perform the following steps to compute the product of their conjugates \(\overline{z} \cdot \overline{w}\):

• Write \(\overline{z} = a - bi\) and \(\overline{w} = c - di\).
• Multiply \((a - bi)(c - di)\) and expand.
• Utilize \(i^2 = -1\) for simplification.
• Obtain the expression \((ac - bd) - (ad + bc)i\).

The beauty of this product is its equivalence to the conjugate of the product of \(z\) and \(w\), demonstrating an important property in complex arithmetic.

Expansion of Expressions

The expansion of expressions, especially in the context of complex numbers, involves multiplying out the components of an expression to simplify it or express it in a more useful form. For instance, when multiplying two complex numbers like \(z = a + bi\) and \(w = c + di\), the expansion is executed as follows:

• Form the expression \((a + bi)(c + di)\).
• Apply the distributive property: \(ac + adi + bci + bdi^2\).
• Simplify using \(i^2 = -1\) to achieve \(ac - bd + (ad + bc)i\).

The expansion allows us to see how individual elements interact, providing insight into how the real and imaginary parts combine and manipulate, which is pivotal in complex number calculations.