Problem 27 - Algebra A Combined Function

Question

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (5-6 i)-4 i $$

Step-by-Step Solution

Answer

The answer is \(5 - 10i\).

Step 1: Identify Like Terms

First, identify the real and imaginary components in the expression \((5-6i)-4i\). Here, the real part is 5 and the imaginary parts are \(-6i\) and \(-4i\).

Step 2: Simplify the Imaginary Terms

Combine the imaginary terms \(-6i\) and \(-4i\) by adding their coefficients. This gives: \(-6 - 4 = -10\). Thus, the new imaginary term is \(-10i\).

Step 3: Combine Real and Imaginary Parts

After simplifying the expression, you have a real part of 5 and an imaginary part of \(-10i\). Combine them to get the final answer in the form \(a + bi\), which is \(5 - 10i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a fascinating part of mathematics. They arise from the need to solve equations that have no real number solutions, such as the square root of a negative number. The basic unit of imaginary numbers is denoted as "i," where \( i \) is defined as \( \sqrt{-1} \). When squared, i becomes -1, such that \( i^2 = -1 \). This simple yet powerful idea allows us to extend the number system beyond just real numbers.

Imaginary numbers are always written in terms of this "i." For instance, \(-6i\) or \(-4i\) in our original exercise represents an imaginary component of a complex number. Combining imaginary numbers, just like real numbers, involves basic arithmetic operations. Imaginary numbers can be used to form complex numbers, which are numbers made up of both a real part and an imaginary part.

Addition of Complex Numbers

Addition of complex numbers involves combining both their real and imaginary components separately. Consider two complex numbers like \( a + bi \) and \( c + di \). To add them, you sum the real parts \( a \, and \, c \) and the imaginary parts \( b \, and \, d \) separately.

The formula for adding these complex numbers looks like this:

• Real parts: \( a + c \)
• Imaginary parts: \( b + d \) which becomes \((a+c) + (b+d)i \)

In our example, there's actually a subtraction operation being performed, but the same principle applies for addition. When simplifying the expression, it's useful to think of like terms just as you do when combining regular algebraic terms.

Subtraction of Complex Numbers

Subtraction of complex numbers is similar to addition, with the key difference being that you subtract the components instead of adding them. Given two complex numbers \( a + bi \) and \( c + di \), you will subtract the real parts and the imaginary parts separately. The operation looks like this:

• Real parts: \( a - c \)
• Imaginary parts: \( b - d \) resulting in \((a-c) + (b-d)i\)

In the exercise provided, you are asked to subtract 4i from \(5 - 6i\). Here, the imaginary components \(-6i\) and \(-4i\) are subtracted, resulting in \(-10i\).

Meanwhile, the real component, which is 5, remains unchanged as there is no other real number to subtract. Combining these gives you the final complex number expressed as \(5 - 10i\). Subtraction, like addition, is very structured - just ensure to manage each part separately and then recombine for your final answer.