Problem 45 - Precalculus Mathematics for Calculus

Question

Evaluate the expression and write the result in the form \(a+b i\) $$\frac{1}{1+i}-\frac{1}{1-i}$$

Step-by-Step Solution

Answer

The result is \(-i\).

Step 1: Write the Expression

The given expression is \(\frac{1}{1+i} - \frac{1}{1-i}\). Our goal is to simplify this expression and express the result in the form \(a+b i\).

Step 2: Rationalize the Denominators

To rationalize the denominators, we'll multiply the numerator and denominator of each fraction by the conjugate of the denominator. \(\frac{1}{1+i} \times \frac{1-i}{1-i} = \frac{1-i}{1+i-1-i^2}\) and \(\frac{1}{1-i} \times \frac{1+i}{1+i} = \frac{1+i}{1-i+i^2}\).

Step 3: Simplify the Denominators

Evaluate the expressions for the denominators by substituting \(i^2 = -1\).- For \(1-i^2 = 1 - (-1) = 2\), the first expression becomes \(\frac{1-i}{2}\).- For \(1+i^2 = 1 - (-1) = 2\), the second expression becomes \(\frac{1+i}{2}\).

Step 4: Simplify the Expression

Now substituting back, we have \(\frac{1-i}{2} - \frac{1+i}{2}\). We can combine the fractions as \(\frac{(1-i)-(1+i)}{2}\).

Step 5: Simplify the Numerator

Now simplify the numerator: \((1-i)-(1+i) = 1 - i - 1 - i = -2i\).

Step 6: Finalize the Calculation

Substitute the simplified numerator into the fraction:\[\frac{-2i}{2} = -i\]. Consequently, the result is \(0 - i\).

Key Concepts

Rationalizing Denominators

When dealing with complex numbers, rationalizing the denominator is often necessary. This means eliminating any imaginary numbers in the denominator. It helps simplify the expression and make it easier to understand and work with.

To rationalize the denominator of a complex number, you multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate is found by changing the sign between the real and imaginary parts. For instance, the conjugate of \(1+i\) is \(1-i\).

The process typically involves:

• Identifying the complex number in the denominator.
• Multiplying the numerator and the denominator by the conjugate of the denominator.
• Simplifying the expression by using the identity \(i^2 = -1\) to remove the imaginary unit from the denominator.

This operation makes the denominator a real number, allowing you to express the final simplified version of your expression accordingly.

Simplifying Expressions

Simplifying expressions, especially those involving complex numbers, is crucial to obtain results in the form \(a+bi\). This involves combining like terms and reducing the expression to the simplest form.

To simplify a complex expression, steps like combining real and imaginary components separately and factoring out common terms are performed. Let's break it down:

• First, ensure the denominators are rationalized if there are fractions involved.
• Simplify each part of the expression individually.
• Combine the simplified parts back into a single expression.

In our example, after rationalizing the denominators, you see that combining \(\frac{1-i}{2}\) and \(\frac{-1-i}{2}\) leads to the cancellation of the real terms, leaving a straightforward calculation. This results in a simplified expression, making calculations and interpretations clearer.

Imaginary Unit

The imaginary unit \(i\) is fundamental in the study of complex numbers, defined as \(i^2 = -1\). This definition creates the imaginary unit, serving as the building block for all complex numbers.

In practical terms, dealing with \(i\) involves:

• Substituting \(i^2\) with \(-1\) wherever it appears in an equation or expression.
• Ensuring that final expressions are left in the form \(a+bi\) where \(a\) and \(b\) are real numbers.
• Understanding that \(i\) enables us to work with roots of negative numbers, expanding our ability to solve algebraic equations beyond those solvable using just real numbers.

In our example, understanding how \(i\) works was key to simplifying the denominator expressions effectively. It eliminates the imaginary parts from them and simplifies the entire expression, making complex arithmetic manageable. Recognizing these principles allows students to handle complex expressions with confidence.