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Chapter 6: Problem 7

Calculate the given expression. \((1+i) /(1-i)\)

Short Answer

Expert verified
The result of the expression \( \frac{1+i}{1-i} \) is \( i \).

Step by step solution

01

- Write the given expression

The given expression is \( \frac{1+i}{1-i} \)
02

- Multiply numerator and denominator by the complex conjugate of the denominator

The complex conjugate of \( 1-i \) is \( 1+i \). Multiply both the numerator and the denominator by \( 1+i \: \) \[ \frac{(1+i) \cdot (1+i)}{(1-i) \cdot (1+i)} \]
03

- Simplify the numerator

Use the distributive property to expand the numerator: \[ (1+i)(1+i) = 1 + i + i + i^2 = 1 + 2i - 1 = 2i \]
04

- Simplify the denominator

Use the difference of squares formula to simplify the denominator: \[ (1-i)(1+i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 \]
05

- Combine the simplified terms

Now, combining the simplified numerator and denominator, we get: \[ \frac{2i}{2} = i \]

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Key Concepts

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

complex conjugate
Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. For example, if you have a complex number like \( 1 - i \), its complex conjugate is \( 1 + i \).
In the earlier exercise, we saw the complex conjugate was used to simplify the fraction. By multiplying both the numerator and denominator by the conjugate of the denominator, we eliminated the imaginary part in the denominator. This technique helps simplify expressions involving complex numbers.
Using complex conjugates effectively makes calculations cleaner and easier to handle, especially when dividing complex numbers.
distributive property
The distributive property allows us to multiply a number by a sum or difference inside parentheses. It states that \(a(b+c) = ab + ac\).
In the given exercise, the distributive property was utilized when expanding \((1+i)(1+i)\):
  • First, \((1+i) \cdot 1 = 1+i\)
  • Then, \((1+i) \cdot i = i + i^2\)
Combining all terms, we get \(1 + i + i + i^2\). Understanding how to distribute terms correctly is crucial for simplifying expressions in algebra and complex numbers.
difference of squares
The difference of squares formula is an identity that states \((a - b)(a + b) = a^2 - b^2\). This formula is particularly useful when dealing with complex numbers.
In our example, we used this property to simplify the denominator \((1 - i)(1 + i)\). Breaking it down:
  • Square the first term: \(1^2 = 1\)
  • Square the second term: \((-i)^2 = -i^2 = 1\)
Combining these, \(1 - (-1) = 1 + 1 = 2\), which gave us a real number in the denominator. Recognizing and using the difference of squares can streamline the process of simplifying complex expressions.

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