Problem 14 - Precalculus

Question

Evaluate each expression using the values \(z=2+3 i, w=9-4 i,\) and \(w_{1}=-7-i\). $$z+\left(w+w_{1}\right)$$

Step-by-Step Solution

Answer

The result is \(4 - 2i\).

Step 1: Evaluate Expression Inside the Parentheses

First, evaluate the expression inside the parentheses, which is \(w + w_1\). Since \(w = 9 - 4i\) and \(w_1 = -7 - i\), perform the addition component-wise:\[(9 - 4i) + (-7 - i) = (9 - 7) + (-4i - i) = 2 - 5i.\]So, \(w + w_1 = 2 - 5i\).

Step 2: Add Result to z

Now that we know \(w + w_1 = 2 - 5i\), we can add this result to \(z\). Given that \(z = 2 + 3i\), perform the addition component-wise:\[(2 + 3i) + (2 - 5i) = (2 + 2) + (3i - 5i) = 4 - 2i.\]

Step 3: Final Result

The final result of the given expression is \(4 - 2i\).

Key Concepts

Complex Number Addition

When dealing with complex numbers, addition is a straightforward yet essential operation. Complex numbers are written in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. In the context of this exercise, we're looking at adding several such numbers together.

To add complex numbers, you simply combine their respective real and imaginary parts separately. So if you have \(z = a + bi\) and another complex number \(w = c + di\), then their sum is \((a + c) + (b + d)i\).

For example, given the complex numbers \(w = 9 - 4i\) and \(w_1 = -7 - i\), you perform the addition as follows:

• Add the real parts: \(9 + (-7) = 2\).
• Add the imaginary parts: \(-4 + (-1) = -5\).

The result is the new complex number \(2 - 5i\). This approach keeps the computation simple and manageable.

Algebraic Operations

Algebraic operations with complex numbers work similarly to those with real numbers, but they involve the additional component of imaginary numbers. This complexity often requires us to handle each part with care.

In this exercise, we're adding three complex numbers. The rule of operations tells us to perform operations inside parentheses first. Thus, we add \(w\) and \(w_1\) first to get \(2 - 5i\). This intermediate result is then added to \(z\).

• Real parts of \(z = 2 + 3i\) and the result \(2 - 5i\) combine to \(2 + 2 = 4\).
• Imaginary parts combine: \(3 - 5 = -2\).

Hence, the sum is \(4 - 2i\). When performing algebraic operations, always keep track of the real and imaginary components separately. It ensures clarity and correctness in your calculations.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a fundamental component in the study of complex numbers. It is defined by the property \(i^2 = -1\), which sets it apart from real numbers and allows complex numbers to represent dimensions beyond the real number line.

This property of \(i\) enables complex numbers to take on a form like \(a + bi\). When performing operations like addition, the imaginary unit operates just as any other variable would in elementary algebra, where it serves to signify components that contribute to solutions in a unique way.

By understanding \(i\) as fundamentally similar but distinct from real numbers, algebra involving complex numbers becomes more intuitive. It allows users to expand the scope of their solutions from the typical real-number restrictions to broader, more complex number planes. Always remember that when combining numbers like \(bi\), you're manipulating that added dimension authorized by \(i\).