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Magazine Chapter 8: Problem 14
\(\frac{3+2 i}{3-2 i}\)
Short Answer
Expert verified
The simplified form is \( \frac{5}{13} + \frac{12}{13}i \).
Step by step solution
01
Identify the Conjugate
To simplify the expression \( \frac{3+2i}{3-2i} \), we first identify the conjugate of the denominator. The conjugate of \(3 - 2i\) is \(3 + 2i\).
02
Multiply Numerator and Denominator by the Conjugate
Next, multiply both the numerator and the denominator by the conjugate \(3 + 2i\). This gives: \[\frac{(3+2i)(3+2i)}{(3-2i)(3+2i)}\]
03
Expand the Expressions
Now, expand both the numerator and the denominator:For the numerator: \((3+2i)(3+2i) = 9 + 6i + 6i + 4i^2 = 9 + 12i - 4 = 5 + 12i\) (since \(i^2 = -1\)).For the denominator:\((3-2i)(3+2i) = 9 - 4i^2 = 9 + 4 = 13\).
04
Simplify the Expression
With the expanded forms, the expression becomes:\[\frac{5 + 12i}{13}\]Separate the real and imaginary parts:\[\frac{5}{13} + \frac{12}{13}i\]
05
Conclusion
The simplified form of \( \frac{3+2i}{3-2i} \) is \( \frac{5}{13} + \frac{12}{13}i \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Conjugate
A conjugate in the context of complex numbers is essential for simplifying expressions involving imaginary units. When we talk about the conjugate of a complex number, we flip the sign of its imaginary part. For a complex number of the form \( a + bi \), the conjugate is \( a - bi \). This operation helps when dividing complex numbers.
Here is why the conjugate is crucial:
Here is why the conjugate is crucial:
- Neutralizes the imaginary part: When we multiply a complex number by its conjugate, the imaginary components cancel out.
- Results in a real number: The product of a complex number and its conjugate gives us a real number. This simplifies division.
Imaginary Unit
The imaginary unit, represented by \(i\), is a fundamental component of complex numbers. It is defined by the property that \(i^2 = -1\). This might seem strange at first, but it's quite useful in various fields of science and engineering.
Key aspects of the imaginary unit:
Key aspects of the imaginary unit:
- Enables the representation of two-dimensional numbers: Complex numbers consist of a real part and an imaginary part, often expressed as \( a + bi \).
- Used in solutions to equations: Some equations don't have real solutions, but they do have complex ones.
Expression Simplification
Expression simplification in complex numbers often involves several steps to reduce a complex expression to its simplest form. Simplifying expressions reveal an expression's essential features and makes it easier to interpret and use.
In this exercise, simplifying was achieved by:
In this exercise, simplifying was achieved by:
- Multiplying by the conjugate: This step transforms the denominator into a real number.
- Expanding terms: By expanding both the numerator and the denominator, we reorganize the terms to combine and eliminate the imaginary units where possible.
- Separating real and imaginary parts: The end goal is to present the expression in the form \( \frac{a}{b} + \frac{c}{b}i \), which shows clarity in the result.
