Problem 74 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Write down the expressions for complex conjugates

The complex conjugate of any complex number is obtained by reversing the sign of its imaginary part. For the given numbers, the complex conjugates are \( \bar{z} = a - bi \) and \( \bar{w} = c - di \).

Step 2: Calculate the product of z and w

The product of \( z = a + bi \) and \( w = c + di \) is calculated using the distributive property: \[ zw = (a + bi)(c + di) = ac + adi + bci + bdi^2 \]By substituting \( i^2 = -1 \), it simplifies to: \[ zw = ac + adi + bci - bd = (ac - bd) + (ad + bc)i \]

Step 3: Calculate the complex conjugate of zw

Now, we need to determine \( \overline{zw} \), which involves reversing the sign of the imaginary part of \( zw \):\[ \overline{zw} = (ac - bd) - (ad + bc)i \]

Step 4: Calculate the product of conjugates \(\bar{z}\) and \(\bar{w}\)

Using the expressions from Step 1, compute the product \( \bar{z} \cdot \bar{w} \):\[ \bar{z} \cdot \bar{w} = (a - bi)(c - di) = ac - adi - bci + bdi^2 \]Since \( i^2 = -1 \), this simplifies to:\[ \bar{z} \cdot \bar{w} = ac + bd - (ad + bc)i \]

Step 5: Verify both expressions are equal

Both expressions \( \overline{zw} = (ac - bd) - (ad + bc)i \) and \( \bar{z} \cdot \bar{w} = (ac - bd) - (ad + bc)i \) are identical after simplification. This proves that \( \overline{zw} = \bar{z} \cdot \bar{w} \).

Key Concepts

Complex Conjugate

A complex conjugate of a complex number is found by changing the sign of its imaginary part. For example, if a complex number is represented as \( z = a + bi \), where \( a \) and \( b \) are real numbers, its complex conjugate is \( \bar{z} = a - bi \).
The importance of complex conjugates lies in their ability to simplify the division and multiplication of complex numbers and provide a real number when multiplied by their original complex number.

• For any complex number \( z \), the product \( z \cdot \bar{z} \) is always a real number: \( z \cdot \bar{z} = (a + bi)(a - bi) = a^2 + b^2 \).
• Complex conjugates are often used to find the modulus or absolute value of a complex number. The modulus of \( z \) is given by \( |z| = \sqrt{z \cdot \bar{z}} = \sqrt{a^2 + b^2} \).
• This property is especially useful in complex number equations and transformations.

Distributive Property

The distributive property allows us to multiply a sum by another number. This property is crucial when dealing with complex numbers, especially when multiplying two complex numbers together. For the expression \( (a + bi)(c + di) \), the distributive property is applied as follows:

• First, distribute \( a \) over \( (c + di) \): \( a(c + di) = ac + adi \).
• Then, distribute \( bi \) over \( (c + di) \): \( bi(c + di) = bci + bdi^2 \). Since \( i^2 = -1 \), this becomes \( bci - bd \).

Combining these terms, we simplify the product to \( zw = (ac - bd) + (ad + bc)i \). This property ensures that no terms are lost in the expansion, making it an essential tool for handling complex arithmetic.

Imaginary Unit

The imaginary unit, denoted as \( i \), is a fundamental component of complex numbers. It is defined by its property \( i^2 = -1 \). This definition is the cornerstone for calculations involving complex numbers, enabling their applications in various fields of mathematics and engineering.

• When multiplying powers of \( i \), the cycle: \( i^0 = 1 \), \( i^1 = i \), \( i^2 = -1 \), \( i^3 = -i \), \( i^4 = 1 \), and so forth, repeats every four terms.
• This cyclic behavior simplifies computations, reducing higher powers of \( i \) to simpler expressions.
• The imaginary unit transforms real numbers into complex numbers, as seen in \( a + bi \), where \( a \) is the real part, and \( bi \) is the imaginary part.

Through the imaginary unit, complex numbers are comprehensible and manageable, making them highly useful in advanced mathematics, physics, and engineering applications.