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Chapter 1: Problem 16

Perform the indicated operations and write your answers in the form \(a+b i,\) where \(a\) and \(b\) are real mombers. $$(6-7 i)-(3-4 i)$$

Short Answer

Expert verified
3-3i

Step by step solution

01

- Identify the Complex Numbers

Recognize the complex numbers in the expression. The given expression is egtr (6-7i)-(3-4i).
02

- Break Down the Expression

Separate the real and imaginary parts of each complex number. For the complex number 6-7i, the real part is 6 and the imaginary part is -7i . Similarly, for the complex number 3-4i , the real part is 3 and the imaginary part is -4i .
03

- Subtract the Real Parts

Subtract the real part of the second complex number from the real part of the first complex number:(6)-(3)=3 .
04

- Subtract the Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first complex number: (-7i)-(4i)= -3i.
05

- Combine the Results

Combine the results of the real and imaginary parts. The result is (3) + (-3i)= 3-3i.

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Key Concepts

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

real part
A complex number is generally represented as \(a + b i\), where \(a\) and \(b\) are real numbers and \( i\) is the imaginary unit with the property that \( i^2 = -1 \).
The real part of a complex number is the 'a' in the expression.
For example, in the expression \(6 - 7i\), the real part is 6.
Likewise, in \(3 - 4i\), the real part is 3.
Identifying the real part is crucial when performing operations like addition and subtraction on complex numbers.
It's simply the part of the complex number that isn鈥檛 multiplied by \(i\).
imaginary part
The imaginary part of a complex number is the coefficient of the imaginary unit \( i \).
In the complex number representation \(a + b i\), the imaginary part is \( b i \).
Here, \( b \) is a real number, but the entire term \( b i \) is called the imaginary part.
For example, the imaginary part of \(6 - 7 i\) is -7i and for \(3 - 4i\), it is -4i.
The imaginary part is important for operations with complex numbers, especially when adding or subtracting them, as it handles the \( i \) component.
subtracting complex numbers
Subtracting complex numbers involves separating the real and imaginary parts, then subtracting them separately.
Here's how:
  • First, identify the real and imaginary parts of each complex number.
  • Next, subtract the real parts from each other.
  • Then, subtract the imaginary parts from each other.
For example, in the problem \((6 - 7i) - (3 - 4i)\):
  • The real parts are 6 and 3, so subtract them to get 3.
  • The imaginary parts are -7i and -4i, so subtract them to get -3i.
Combining these results, we get the final answer 3 - 3i.
complex number operations
Complex number operations include addition, subtraction, multiplication, and division.
Each operation has specific rules:
  • Addition: Sum the real parts and the imaginary parts separately.
  • Subtraction: Subtract the real parts and the imaginary parts separately.
  • Multiplication: Expand using the distributive property and simplify, keeping in mind that \(i^2 = -1\).
  • Division: Multiply the numerator and denominator by the conjugate of the denominator, then simplify.
Mastering these basic operations is crucial in understanding and working with complex numbers.

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