Problem 11 - Precalculus

Question

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

Step-by-Step Solution

Answer

(a) 11 - i; (b) 11 - 7i; (c) 4.

Step 1: Add Complex Numbers for part (a)

We have the expression \(z+w\), where \(z=2+3i\) and \(w=9-4i\). To add these complex numbers, add their real parts and their imaginary parts separately: \[ z + w = (2+9) + (3i - 4i) = 11 - i. \]

Step 2: Add Conjugate for part (b)

In part (b), the expression is \(\bar{z} + w\), where \(\bar{z}\) represents the conjugate of \(z=2+3i\). The conjugate is \(\bar{z} = 2-3i\). Now add \(\bar{z}\) and \(w=9-4i\): \[ \bar{z} + w = (2-3i) + (9-4i) = (2+9) + (-3i - 4i) = 11 - 7i. \]

Step 3: Calculate Adding with Conjugate for part (c)

In part (c), the expression is \(z + \bar{z}\). Since \(z = 2 + 3i\), its conjugate is \(\bar{z} = 2 - 3i\). Add them together: \[ z + \bar{z} = (2 + 3i) + (2 - 3i) = (2+2) + (3i-3i) = 4. \]

Key Concepts

Adding Complex Numbers

Adding complex numbers is a basic operation that involves combining two or more complex numbers to produce a new complex value. Each complex number has a real part and an imaginary part, typically written in the form of a + bi, where:

• a is the real part
• b is the imaginary part

To add complex numbers,

• You sum their real parts together.
• Then, you sum their imaginary parts separately.

For example, if you have two complex numbers, z = 2 + 3i and w = 9 - 4i, you add them by: First, adding the real parts: 2 + 9 = 11.
Then, adding the imaginary parts: 3i - 4i = -i.
So, the result of z + w is 11 - i. This approach simplifies complex number addition by addressing each component of the numbers separately, ensuring clarity and specificity in mathematical operations.

Complex Conjugate

The concept of a complex conjugate is central to the manipulation of complex numbers. A complex conjugate of a complex number is obtained by changing the sign of its imaginary part. If the complex number is written as a + bi, then its complex conjugate is a - bi.
The complex conjugate serves important roles, especially in simplifying division of complex numbers and solving polynomial equations. Additionally, conjugates are used to find the magnitude or absolute value of a complex number.Using the example provided in the exercise, where z = 2 + 3i, the complex conjugate of z would be \(\bar{z} = 2 - 3i\).
In the case where you need to compute \(\bar{z} + w\) with \(w = 9 - 4i\), you add the conjugate of z to w by first adding the real parts, resulting in 11, and then adding the imaginary parts, resulting in -7i. Hence, the expression evaluates to 11 - 7i. This operation leverages the definition of conjugates to effectively recalibrate the imaginary components of complex addition.

Real and Imaginary Parts

Understanding the real and imaginary parts of a complex number is crucial for effectively performing complex number operations. Each complex number is composed of a real part and an imaginary part. The real part of a complex number a + bi is 'a', while 'b' is the coefficient of the imaginary part 'i'.

For example, in the complex number z = 2 + 3i:

• The real part is 2.
• The imaginary part is 3.

When adding a complex number to its conjugate, such as z + \(\bar{z}\), it always results in a real number. This is because the imaginary parts cancel each other out. Given that z = 2 + 3i, and its conjugate \(\bar{z} = 2 - 3i\), summing them yields:0 imaginary component: \(3i - 3i = 0\).
Real component: \(2 + 2 = 4\).
Thus, complex operations like these underline the distinct roles of real and imaginary parts in arithmetic calculations involving complex numbers. Having a strong grasp of these aspects enables successful manipulation and interpretation of complex expressions.