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Magazine Chapter 9: Problem 112
Perform the operations. Write all answers in the form \(a+b i .\) $$ \frac{5-i}{3+2 i} $$
Short Answer
Expert verified
The simplified form in standard form is \(1-i\).
Step by step solution
01
Understand the Task
We need to simplify the expression \( \frac{5-i}{3+2i} \) and write it in the form \( a+bi \), where \( a \) and \( b \) are real numbers.
02
Determine the Conjugate
To simplify the expression, we will multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 3+2i \) is \( 3-2i \).
03
Multiply Numerator and Denominator by the Conjugate
Multiply both the numerator \((5-i)\) and the denominator \((3+2i)\) by the conjugate \((3-2i)\).\[\frac{(5-i)(3-2i)}{(3+2i)(3-2i)}\]
04
Calculate the Numerator
Expand the product in the numerator:\[(5-i)(3-2i) = 5\cdot3 + 5\cdot(-2i) + (-i)\cdot3 + (-i)\cdot(-2i)\]\[= 15 - 10i - 3i + 2i^2\]Since \(i^2 = -1\), substitute to get:\[15 - 13i + 2(-1) = 15 - 13i - 2 = 13 - 13i\]
05
Calculate the Denominator
Expand the product in the denominator:\[(3+2i)(3-2i) = 3^2 - (2i)^2\]\[= 9 - 4i^2\]Since \(i^2 = -1\), substitute to get:\[9 - 4(-1) = 9 + 4 = 13\]
06
Write the Result in Standard Form
Now we have:\[\frac{13-13i}{13} = \frac{13}{13} - \frac{13i}{13}\]Simplifying, we get:\[1 - i\]
07
Verify the Form
Check that the result \(1-i\) is in the required form \(a + bi\). It is, with \(a = 1\) and \(b = -1\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conjugate of a Complex Number
Complex numbers have two components: a real part and an imaginary part. The conjugate of a complex number helps simplify and perform complex number calculations. If you have a complex number like \(3 + 2i\), its conjugate is formed by changing the sign of the imaginary part, resulting in \(3 - 2i\). This is very useful in division operations involving complex numbers.
Why do we care about conjugates? Great question! Here鈥檚 why:
Why do we care about conjugates? Great question! Here鈥檚 why:
- They help eliminate the imaginary part in the denominator when dividing complex numbers, making calculations smoother.
- Multiplying a complex number by its conjugate results in a real number. For example, \((3 + 2i)(3 - 2i) = 9 + 4 = 13\).
- Conjugates also maintain the modulus of a complex number, meaning the length or magnitude remains the same.
Imaginary Unit
The imaginary unit is denoted by \(i\), and it is defined by the property \(i^2 = -1\). This imaginary unit is what allows complex numbers to exist and interact in unique ways that regular numbers cannot.
Here鈥檚 a little more about this fascinating concept:
Here鈥檚 a little more about this fascinating concept:
- When you see \(i\), think of it as the square root of -1. Normally, there is no real number solution to a square root of a negative number, but \(i\) fills that gap.
- \(i\) on its own is called purely imaginary because it stands without a real part until combined with other numbers.
- It鈥檚 essential in calculations because it changes signs when squared, making it crucial in complex number arithmetic.
Standard Form
The standard form of a complex number is expressed as \(a + bi\). Here, \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
Standard form is important because:
Standard form is important because:
- It offers a clear representation of the complex number, separating the real part from the imaginary part.
- Having all complex numbers in this form allows for easy comparison and computation.
- Operations like addition, subtraction, and multiplication are straightforward when numbers are in standard form.
