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Chapter 8: Problem 34

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (9+i)-(3+2 i) $$

Short Answer

Expert verified
6 - i

Step by step solution

01

Identify the Real and Imaginary Parts

Consider the given expression: (9+i)-(3+2i). Identify the real and imaginary parts of each complex number. The first complex number: 9 (real part) and 1 (imaginary part). The second complex number: 3 (real part) and 2 (imaginary part).
02

Subtract the Real Parts

Subtract the real part of the second complex number from the real part of the first complex number: 9 - 3 = 6
03

Subtract the Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first complex number: 1 - 2 = -1i
04

Combine the Results

Combine the results from Step 2 and Step 3 to form the final answer: 6 + (-1)i This simplifies to 6 - i

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Key Concepts

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

Complex Numbers
Complex numbers are numbers that have both a real part and an imaginary part. Imagine them as two-dimensional numbers, where one part represents a real number and the other part represents an imaginary number. The imaginary part is usually denoted by 'i', where i is the square root of -1.
For example, in the complex number 9+i, 9 is the real part and i is the imaginary part.
Real Part
The real part of a complex number is the component that does not involve 'i'. In the expression (9+i)-(3+2i), the real part of 9+i is 9 and the real part of 3+2i is 3. When subtracting complex numbers, you first subtract the real parts from each other.
In this case, 9 (real part of the first number) minus 3 (real part of the second number) equals 6.
Imaginary Part
The imaginary part of a complex number is the component that involves 'i'. It indicates the 'imaginary' dimension of the number. In (9+i)-(3+2i), the imaginary part of 9+i is 1 (because i is equivalent to 1i) and the imaginary part of 3+2i is 2i.
To subtract the imaginary parts, you perform the same operation on the coefficients of i. Here, 1 (imaginary part of the first number) minus 2 (imaginary part of the second number) equals -1i.
Subtraction
To subtract complex numbers, you treat the real parts and imaginary parts separately. Subtract the real parts together and the imaginary parts together, and then combine the results. Our example is (9+i)-(3+2i).
Step 1: Subtract the real parts: 9 - 3 = 6
Step 2: Subtract the imaginary parts: 1 - 2 = -1i
Step 3: Combine the results from Step 1 and Step 2: 6 + (-1)i or simply 6 - i.
Thus, the result of the subtraction is a new complex number: 6 - i.

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