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Chapter 6: Problem 6

Calculate the given expression. \(\overline{(2+3 i)}-\overline{(-3-2 i)}\)

Short Answer

Expert verified
5 - 5i

Step by step solution

01

- Understand the problem

The problem requires calculating the expression \( \overline{(2+3i)} - \overline{(-3-2i)} \). This involves finding the conjugates of the given complex numbers and then subtracting them.
02

- Find the conjugate of (2+3i)

The conjugate of a complex number \(a + bi\) is \(a - bi\). Here, the conjugate of \(2 + 3i\) is \(2 - 3i\).
03

- Find the conjugate of (-3-2i)

Similarly, the conjugate of \(-3 - 2i\) is \(-3 + 2i\).
04

- Subtract the conjugates

Now, subtract the conjugate of \(-3 - 2i\) from the conjugate of \(2 + 3i\): \( (2 - 3i) - (-3 + 2i) \). This simplifies to: \ 2 - 3i + 3 - 2i = (2 + 3) + (-3i - 2i) = 5 - 5i \.

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Key Concepts

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

complex conjugates
When dealing with complex numbers, one useful concept is the complex conjugate. A complex number is generally written as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. The complex conjugate of \(a + bi\) is \(a - bi\).

To find the complex conjugate of a number, you simply change the sign of the imaginary part. For example, the conjugate of the complex number \(2 + 3i\) is \(2 - 3i\).

This concept helps to simplify various operations involving complex numbers, especially when wanting to eliminate imaginary parts in denominators or during subtraction tasks like the one in our exercise.
subtraction of complex numbers
Subtraction is one of the basic operations you can perform on complex numbers. It is similar to subtracting polynomials. To subtract two complex numbers \(a + bi\) and \(c + di\), you subtract their real parts and their imaginary parts separately.

Mathematically, this is written as:
\((a + bi) - (c + di) = (a - c) + (b - d)i\)
For example, in our exercise, we have the complex numbers \(2 - 3i\) and \(-3 + 2i\).
We perform the subtraction as follows:
  • Subtract the real parts: \ 2 - (-3) = 2 + 3 = 5 \
  • Subtract the imaginary parts: \ -3i - 2i = -5i \
Therefore, the result of \( (2 - 3i) - (-3 + 2i) = 5 - 5i \).
elementary algebra
Elementary algebra is the foundation for most math operations, including those involving complex numbers. It covers the basic rules for operations such as addition, subtraction, and finding conjugates, all of which we used in our given exercise.

Here are some key points to remember:
  • Perform operations separately for real and imaginary parts when dealing with complex numbers.

  • If the question involves conjugates, remember to change the sign of the imaginary part.

  • Practice makes perfect: the more you work with these kinds of problems, the more intuitive the processes will become.

Understanding these basic algebraic rules makes solving complex number problems much easier and allows for more straightforward computations, as demonstrated in our step-by-step solution.

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