Problem 23 - Algebra A Combined Function

Question

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ 6-(8+4 i) $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Expression

The problem provided is to subtract the complex number \((8 + 4i)\) from 6. The expression is written as \(6 - (8 + 4i)\).

Step 2: Distribute the Negative Sign

To solve the expression, distribute the negative sign to both terms inside the parenthesis:\[6 - (8 + 4i) = 6 - 8 - 4i\]

Step 3: Combine the Real Parts

Now, combine the real numbers (non-complex part):\[6 - 8 = -2\]

Step 4: Write the Imaginary Part

The imaginary part is still \(-4i\) because there was no other imaginary number to combine it with. So we have:\[-2 - 4i\]

Step 5: Conclude in the Form \(a+bi\)

The final result is in the form \(a + bi\), where \(a = -2\) and \(b = -4\): \(-2 - 4i\)

Key Concepts

Real Part

When dealing with complex numbers, it's essential to understand the structure of these numbers. Each complex number can be expressed in the form \(a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part. The real part, \(a\), is simply a regular real number, untouched by any imaginary units like \(i\).

The real part is crucial because it helps us understand how the complex number behaves on the real axis of the complex plane. In the exercise, the complex number \((8 + 4i)\) features a real part of 8. However, we are subtracting it from 6, not another complex number. The result of this operation is found by performing the arithmetic solely on the real parts:

• Real part of 6 is 6
• Real part of \((8 + 4i)\) is 8

By subtracting, we combine these parts: \(6 - 8 = -2\). Thus the real part of the resulting complex number is \(-2\).

Imaginary Part

The imaginary part of a complex number is paired with the imaginary unit \(i\). For a complex number \(a + bi\), \(b\) denotes the imaginary part. This part illustrates how the number extends along the imaginary axis on the complex plane. This is essential for performing arithmetic operations on complex numbers, like addition and subtraction.

In the exercise example, the complex number \((8 + 4i)\) has an imaginary part of 4. When subtracting \((8 + 4i)\) from 6, the process involves separating the real and imaginary components:

• The imaginary part \(4i\) from \((8 + 4i)\) becomes \(-4i\) when distributed with a negative sign.

The number 6 does not have an imaginary part (or it's 0 if we consider placeholder terms), so the imaginary part simply remains \(-4i\) in the final expression, contributing to the result of \(-2 - 4i\).

Subtracting Complex Numbers

Subtracting complex numbers involves handling both their real and imaginary components separately. When facing an expression like \(6 - (8 + 4i)\), it's key to distribute any numbers or signs before tackling the subtraction.

Here's the step-by-step process:

• Distribute the negative sign: Change the sign of each part of the number inside the parentheses. For instance, \((8 + 4i)\) when distributed becomes \(-8 - 4i\).
• Real numbers: Subtract the real parts: \(6 - 8 = -2\).
• Imaginary parts: Since 6 has no imaginary part, simply add \(-4i\): \(-2 - 4i\).

By following these steps, the complete subtraction yields the result in the form \(a + bi\). The final answer, \(-2 - 4i\), shows clearly separated real and imaginary parts.