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Chapter 10: Problem 23

Add or subtract. Write the sum or difference in the form \(a+b.\) $$ 6-(8+4 i) $$

Short Answer

Expert verified
The difference is \(-2 - 4i\).

Step by step solution

01

Distribute the Minus Sign

Before we perform operations inside the parentheses, we need to distribute the minus sign across the terms inside. The expression inside the parentheses is \(8 + 4i\), so when we distribute the minus sign, each term has its sign changed: \(-(8 + 4i) = -8 - 4i\).
02

Perform the Subtraction

Now replace \(8 + 4i\) with \(-8 - 4i\) in the original expression. The expression becomes \(6 - 8 - 4i\).
03

Simplify the Expression

Combine like terms in the expression \(6 - 8 - 4i\). The real parts \(6 - 8\) simplify to \(-2\), leaving the imaginary part \(-4i\) unchanged. The expression is now \(-2 - 4i\).
04

Express in the Form \(a + bi\)

Write the expression \(-2 - 4i\) explicitly in the form \(a + bi\). Here, \(a = -2\) and \(b = -4\), so the expression is \(-2 + (-4)i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Subtracting Complex Numbers
Complex numbers are a blend of real numbers and imaginary numbers. A complex number usually takes the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit, which equals the square root of -1.
When we talk about subtracting complex numbers, we are essentially subtracting their real parts and their imaginary parts separately. For example, if we have \((a + bi) - (c + di)\), the result will be \((a-c) + (b-d)i\).
  • Real parts: Subtract the real part of the second complex number from the real part of the first.
  • Imaginary parts: Subtract the imaginary part of the second complex number from the imaginary part of the first.
Always remember to treat real and imaginary parts separately, especially if there are negative signs involved, to avoid mistakes.
Distributing Minus Sign
Distributing a minus sign is a common step in algebra, and it's especially crucial in the context of complex numbers. When you have an expression like \(-(a + bi)\), the minus sign affects both the real part \(a\) and the imaginary part \(bi\).
To effectively distribute the minus sign, follow these steps:
  • Change the sign of the real part: from \(a\) to \(-a\).
  • Change the sign of the imaginary part: from \(bi\) to \(-bi\).
An example is \(-(8 + 4i)\), which transforms into \(-8 - 4i\). Remember, distributing the minus sign properly is key to successfully simplifying the expression.
Simplifying Complex Expressions
Simplifying complex expressions involves combining like terms to make the expression as straightforward as possible.
When dealing with a complex expression, your task is to:
  • Combine the real numbers.
  • Combine the imaginary numbers.
For instance, in an expression like \(6 - 8 - 4i\), first, simplify the real parts by performing the arithmetic operation \(6 - 8 = -2\). Then, simply attach the imaginary part \(-4i\) as it is, resulting in \(-2 - 4i\).
This approach not only makes the expression look cleaner but also helps in understanding and interpreting complex numbers effectively.

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