Problem 45 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ \frac{1}{1+i}-\frac{1}{1-i} $$

Step-by-Step Solution

Answer

The expression evaluates to \( -i \).

Step 1: Find the Conjugates

To simplify each term in the expression, multiply the numerator and the denominator by the conjugate of the denominator. For \( \frac{1}{1+i} \), the conjugate is \( 1-i \), and for \( \frac{1}{1-i} \), it is \( 1+i \).

Step 2: Simplify the First Fraction

Multiply the numerator and denominator of \( \frac{1}{1+i} \) by \( 1-i \):\[\frac{1 \cdot (1-i)}{(1+i)(1-i)} = \frac{1-i}{1^2-i^2} = \frac{1-i}{1-(-1)} = \frac{1-i}{2} = \frac{1}{2} - \frac{i}{2}.\]

Step 3: Simplify the Second Fraction

Multiply the numerator and denominator of \( \frac{1}{1-i} \) by \( 1+i \):\[\frac{1 \cdot (1+i)}{(1-i)(1+i)} = \frac{1+i}{1^2-i^2} = \frac{1+i}{1-(-1)} = \frac{1+i}{2} = \frac{1}{2} + \frac{i}{2}.\]

Step 4: Subtract the Two Simplified Fractions

Subtract the simplified fractions from Step 2 and Step 3:\[\left( \frac{1}{2} - \frac{i}{2} \right) - \left( \frac{1}{2} + \frac{i}{2} \right).\]Combine like terms:\[\frac{1}{2} - \frac{1}{2} = 0 \-\frac{i}{2} - \frac{i}{2} = -i \\Longrightarrow 0 - i = -i.\]

Step 5: Express the Answer in the Form a + bi

Since the real part of the result is \( 0 \) and the imaginary part is \( -1 \), the expression \(-i\) can be written as \( 0 - i\). Therefore, \( a = 0 \) and \( b = -1 \).

Key Concepts

Conjugates

Conjugates play a crucial role in simplifying expressions involving complex numbers. The conjugate of a complex number has the same real part but an opposite sign for the imaginary part. For example, for the complex number \( a + bi \), its conjugate is \( a - bi \). The utility of conjugates comes into play when we need to simplify complex fractions—such as when the denominator contains an imaginary number.

By multiplying both the numerator and denominator of a fraction by the conjugate of the denominator, we can eliminate the imaginary component in the denominator. This process helps to simplify the expression to a more manageable form. In our exercise, multiplying \( \frac{1}{1+i} \) by \( 1-i \) and \( \frac{1}{1-i} \) by \( 1+i \) removed the imaginary numbers from the denominators. This approach simplifies further calculations and transforms the expression into familiar, more easily interpreted forms.

Simplifying Expressions

Simplifying complex expressions, especially those involving fractions, requires some basic algebraic manipulation and knowledge of complex numbers. Simplifying involves steps like multiplying the numerator and the denominator by the conjugate of the denominator, as seen in the original exercise. This operation reduces a complex denominator to a real number, making the expression simpler.

Specific steps include multiplying the fractions and using the formula \( (a+bi)(a-bi) = a^2-b^2i^2 \). This formula helps effectively remove imaginary parts in the denominator since \( i^2 = -1 \). Once reduced, the expression transforms to a more manageable form consisting of real numbers and fractions of imaginary numbers, simplifying subtraction or addition of values in expressions.

Imaginary Numbers

Imaginary numbers form the basis of complex number calculation. The imaginary unit \( i \) is defined as \( \sqrt{-1} \). Thus, any imaginary number can be expressed as a real coefficient multiplied by \( i \), such as \( bi \). Understanding imaginary numbers helps to perform operations with complex numbers by separating real and imaginary parts.

In mathematical expressions, keeping real and imaginary parts separate is key. When simplifying or adding and subtracting complex numbers, always combine like terms: real with real, and imaginary with imaginary. This simplification is demonstrated in our solution, where the subtraction of fractions results in an expression combining real coefficients with imaginary units, ultimately leading to a final expression purely in terms of \( i \).

Fractions with Complex Numbers

Working with fractions that have complex numbers requires a systematic approach to maximize clarity and simplicity. The challenge largely lies in the denominator when it contains a complex number. The standard technique, as shown in the exercise, involves removing the complex component by multiplying by the conjugate. The result is then a real number denominator, allowing for simplified arithmetic operations.

The process demonstrates another layer of fractions with complex numbers: keeping track of the imaginary and real components throughout the operation. This ensures the final result remains in the standard form \( a + bi \). For example, \( \frac{1}{1+i} \) simplified to \( \frac{1}{2} - \frac{i}{2} \), and similarly for the other fraction, keeps these components clear for straightforward movie to subtraction in the last step of the problem. When handling complex fractions, always simplify each part to make calculation easier and more interpretable.