Problem 13 - Precalculus

Question

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

Step-by-Step Solution

Answer

The expression evaluates to \(4 - 2i\).

Step 1: Substitute the Values

First, we need to substitute the given expressions for \(z\), \(w\), and \(w_1\) into the equation. The expression we need to evaluate is \((z + w) + w_1\). Substitute the values: \(z = 2 + 3i\), \(w = 9 - 4i\), and \(w_1 = -7 - i\) into the expression.

Step 2: Calculate \(z + w\)

Calculate \(z + w\) by adding the corresponding real and imaginary parts of \(z\) and \(w\). \(z = 2 + 3i\) and \(w = 9 - 4i\).Real parts: \(2 + 9 = 11\).Imaginary parts: \(3i - 4i = -i\).So, \(z + w = 11 - i\).

Step 3: Add \(w_1\) to \(z + w\)

Now, add \(w_1 = -7 - i\) to the result from Step 2, \(z + w = 11 - i\).Real parts: \(11 - 7 = 4\).Imaginary parts: \(-i - i = -2i\).Therefore, \((z + w) + w_1 = 4 - 2i\).

Key Concepts

Complex Arithmetic

Complex arithmetic refers to performing mathematical operations such as addition, subtraction, multiplication, and division on complex numbers. A complex number is expressed in the form of \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part, with \(b\) being a real number and \(i\) representing the imaginary unit, equal to \(\sqrt{-1}\).
To perform complex arithmetic, it's essential to treat \(i\) (the imaginary unit) with care, especially since its square, \(i^2\), equals \(-1\).
When tackling calculations involving complex numbers, remember:

• Add or subtract like terms specifically, handle the real parts independently from the imaginary parts.
• Use the rule \(i^2 = -1\) to simplify expressions involving higher powers of \(i\).

Understanding and mastering complex arithmetic allows you to solve more advanced problems that incorporate both real and imaginary components.

Imaginary Numbers

An imaginary number is a number that can be written as a real number multiplied by the imaginary unit \(i\). The imaginary unit \(i\) is defined by the property \(i^2 = -1\). Imaginary numbers extend our number system to solve equations that do not have real number solutions.
For example, the equation \(x^2 + 1 = 0\) has no real solution since adding 1 to any real number squared does not give zero. However, with imaginary numbers, the solutions are \(x = i\) and \(x = -i\).
Here are some key points about imaginary numbers:

• They arise in many advanced mathematics fields like engineering, quantum physics, and applied mathematics.
• Paired with real numbers, they form complex numbers, allowing the solution of polynomial equations.
• Imaginary numbers make it easier to perform rotations and oscillations calculations in two-dimensional space.

In computations, understanding imaginary numbers helps interpret the results of equations that otherwise would seem unsolvable in the realm of real numbers.

Addition of Complex Numbers

The addition of complex numbers involves summing their real and imaginary parts separately. Suppose you have two complex numbers: \(z_1 = a + bi\) and \(z_2 = c + di\). The addition is performed as follows: \[ z_1 + z_2 = (a + c) + (b + d)i \]
This process involves two main steps:

• Add the real parts together: Add \(a\) and \(c\) to get the resulting real part.
• Add the imaginary parts together: Add \(b\) and \(d\) to get the resulting imaginary part.

For example, consider adding \(z = 2 + 3i\) and \(w = 9 - 4i\):1. Real parts: \(2 + 9 = 11\)2. Imaginary parts: \(3i - 4i = -i\)Thus, \(z + w = 11 - i\).
The simplicity of addition makes it foundational in handling more complex problems, as shown in the solution to the exercise where further combination involved another complex addition step to find the final result.