Problem 12 - Precalculus

Question

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). (a) \(\frac{\bar{z}+\bar{w}}{(z+w)}\) (b) \(\overline{(z+w)}\) (c) \(w-\bar{w}\)

Step-by-Step Solution

Answer

(a) \(1 + \frac{11}{61}i\), (b) \(11 + i\), (c) \(-8i\)

Step 1: Calculate Conjugates

Firstly, calculate the conjugates of the given complex numbers. For \( z = 2 + 3i \), the conjugate \( \bar{z} \) is \( 2 - 3i \). For \( w = 9 - 4i \), the conjugate \( \bar{w} \) is \( 9 + 4i \).

Step 2: Express the Numerator and Denominator in (a)

For expression (a), calculate \( \bar{z} + \bar{w} \) to get \((2 - 3i) + (9 + 4i) = 11 + i\). Then, calculate \( z + w \) to get \((2 + 3i) + (9 - 4i) = 11 - i \).

Step 3: Evaluate Expression (a)

Now, divide the two results: \[\frac{11 + i}{11 - i}\]. To simplify, multiply numerator and denominator by the conjugate of the denominator: \[\frac{(11 + i)(11 + i)}{(11 - i)(11 + i)}\]. This results in \[\frac{121 + 22i + i^2}{(11)^2 - i^2} = \frac{120 + 22i}{122} = 1 + \frac{11}{61}i\].

Step 4: Calculate Conjugate of (z+w) for (b)

For expression (b), use the sum calculated in Step 2: \( z + w = 11 - i \). The conjugate \( \overline{(z + w)} \) is \( 11 + i \).

Step 5: Evaluate Expression (c)

For expression (c), calculate \( w - \bar{w} \): \( (9 - 4i) - (9 + 4i) = -8i \).

Key Concepts

Complex Conjugate

Complex conjugates are an essential concept in complex numbers. They are helpful in various arithmetic operations and simplifications. A complex number is in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit with the property that \(i^2 = -1\). The complex conjugate of such a number is \(a - bi\).

By changing the sign of the imaginary part, complex conjugates help in rationalizing denominators during division and in simplifying expressions involving complex numbers. For example, for the complex number \(z = 2 + 3i\), its conjugate is \( \bar{z} = 2 - 3i\).

Remember that when multiplying a complex number by its conjugate, you get a real number. For \(z = 2 + 3i\), \(z \times \bar{z} = (2 + 3i)(2 - 3i) = 4 + 9 = 13\). This property is particularly useful when dividing complex numbers.

Addition of Complex Numbers

Adding complex numbers is straightforward. It involves adding the real parts and the imaginary parts separately. If you have two complex numbers, \(z_1 = a + bi\) and \(z_2 = c + di\), their sum is \((a + c) + (b + d)i\).

For example, in the exercise, we add \(z = 2 + 3i\) and \(w = 9 - 4i\):

• Real parts: \(2 + 9 = 11\)
• Imaginary parts: \(3i - 4i = -i\)

This results in \(z + w = 11 - i\), demonstrating that you handle addition of complex numbers much like handling vectors in mathematics.

Division of Complex Numbers

Dividing complex numbers is slightly more complex than addition or subtraction. The goal is to convert the division into a simpler form. This is done by multiplying the numerator and the denominator by the conjugate of the denominator.

Suppose we need to divide \(z_1 = x + yi\) by \(z_2 = u + vi\). We multiply both the numerator and the denominator by the conjugate of \(z_2\), which is \(u - vi\):\[\frac{z_1}{z_2} = \frac{(x + yi)(u - vi)}{(u + vi)(u - vi)}\]

The denominator simplifies to a real number \(u^2 + v^2\), canceling out the imaginary part. The numerator becomes a new complex number after applying the distributive property.

• In the original exercise, \(\frac{11 + i}{11 - i}\) was evaluated by multiplying both parts by \(11 + i\). The denominator \((11 - i)(11 + i)\) simplifies to \(122\).
• This process not only simplifies the division but also ensures the quotient is expressed in standard complex form: a real part plus an imaginary part.

This technique is a go-to method for simplifying fractions involving complex numbers.