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Chapter 7: Problem 44

Add or subtract as indicated. Give answers in standard form. $$(9+i)-(3+2 i)$$

Short Answer

Expert verified
6 - i

Step by step solution

01

Identify the Real and Imaginary Parts

Identify the real and imaginary parts of each complex number. In this case, for the first complex number, the real part is 9 and the imaginary part is i. For the second complex number, the real part is 3 and the imaginary part is 2i.
02

Subtract the Real Parts

Subtract the real parts of the complex numbers: 9 - 3 = 6.
03

Subtract the Imaginary Parts

Subtract the imaginary parts of the complex numbers: i - 2i = -i.
04

Combine the Results

Combine the results from the previous two steps to get the complex number in standard form: 6 - i.

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Key Concepts

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

Subtracting Complex Numbers
Subtracting complex numbers is quite similar to subtracting polynomials. When you are given two complex numbers, a + bi and c + di, you subtract the real parts and the imaginary parts separately.

For example, let's consider the complex numbers (9 + i) and (3 + 2i).
  • First, identify the real and imaginary parts of each number. The real part of 9 + i is 9, and its imaginary part is i. For 3 + 2i, the real part is 3, and the imaginary part is 2i.
  • Then, subtract the real parts: 9 - 3 = 6.
  • Next, subtract the imaginary parts: i - 2i = -i.
  • Finally, combine these results to get the complex number in standard form: 6 - i.

This is how we subtract complex numbers. It's straightforward when you break it into steps like: real part - real part, and imaginary part - imaginary part.
Real and Imaginary Parts
Understanding the real and imaginary parts of a complex number is crucial. Complex numbers are expressed in the form a + bi, where:
  • a is the real part
  • bi is the imaginary part.

In our example, (9 + i) and (3 + 2i), you can easily pick out the real and imaginary components:
  • For 9 + i, the real part is 9, and the imaginary part is i (remember, i is an imaginary unit).
  • For 3 + 2i, the real part is 3, and the imaginary part is 2i.

Breaking this down helps you perform operations like addition, subtraction, multiplication, and division with ease. You simply treat the real and imaginary components separately in calculations.
Standard Form
The standard form of a complex number is written as a + bi. This format makes it easy to see the real and imaginary parts.

In our example, after subtracting (9 + i) - (3 + 2i), we arrived at 6 - i.
  • In this result, 6 is the real part.
  • -i is the imaginary part.

Whenever you find the result of operations involving complex numbers, make sure your answer is in standard form. This format is essential for clarity and makes further calculations simpler.

By always expressing your final answer in the standard form a + bi, you ensure anyone else reading or using your result will easily understand the real and imaginary components.

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