Problem 21 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (-12+8 i)-(7+4 i) $$

Step-by-Step Solution

Answer

-19 + 4i

Step 1: Identify terms in the expression

We are given the expression \[(-12 + 8i) - (7 + 4i)\]Let's identify the real and imaginary parts for both complex numbers. For \(-12 + 8i\), the real part is \(-12\) and the imaginary part is \(8i\). For \(7 + 4i\), the real part is \(7\) and the imaginary part is \(4i\).

Step 2: Subtract the real parts

Now, subtract the real parts of the given complex numbers:\[-12 - 7 = -19\]

Step 3: Subtract the imaginary parts

Next, subtract the imaginary parts of the complex numbers:\[8i - 4i = 4i\]

Step 4: Combine the results

Combine the results from Step 2 and Step 3. The final expression is:\[-19 + 4i\]

Key Concepts

Real Part

When discussing complex numbers, it's essential to understand their composition. A complex number is expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The term \(a\) is known as the "real part." This component behaves just like any other number on the real number line. It neither involves the square root of negative one, referred to as \(i\), nor any imaginary component.

Identifying the real part of a complex number is straightforward. For example, in the complex number \(-12 + 8i\), the real part is \(-12\). Similarly, for \(7 + 4i\), the real part is \(7\).

• The real part doesn't change if you move along the imaginary axis.
• It directly represents the position on the horizontal axis of the complex plane.

Grasping the real part's role is crucial for operations like addition or subtraction concerning complex numbers.

Imaginary Part

In any complex number \(a + bi\), the term \(bi\) represents the 'imaginary part.' Here, \(b\) is a real number, and \(i\) is the imaginary unit, defined as the square root of \(-1\).

The imaginary part allows complex numbers to capture information beyond the traditional number line, extending them into a plane. For the complex number \(-12 + 8i\), the imaginary part is \(8i\). Similarly, in \(7 + 4i\), the imaginary part is \(4i\).

• The imaginary part lies along the vertical axis in the complex plane.
• It provides a means to measure the "height" of a complex number above or below the real line.

Understanding the imaginary part is vital, especially when you perform operations like subtraction or addition between complex numbers.

Subtract Complex Numbers

Subtracting complex numbers involves dealing with both their real and imaginary components separately. The goal is to form a new complex number. Let's walk through subtracting the complex numbers \((-12 + 8i)\) and \((7 + 4i)\).

First, subtract the real parts. So, you take the real part of the first complex number \(-12\) and subtract the real part of the second complex number, \(7\), resulting in:

• \(-12 - 7 = -19\)

Next, subtract the imaginary parts. Likewise, subtract the imaginary part of the second complex number from the imaginary part of the first one:

• \(8i - 4i = 4i\)

Finally, combine these results to form the new complex number. This gives you:

• \(-19 + 4i\)

This method is essential for accurately performing arithmetic with complex numbers, ensuring the calculations maintain both their real and imaginary components effectively.