Problem 37 - Algebra and Trigonometry

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 \(-i\) or \(0 - i\).

Step 1: Simplify the First Term

To simplify \( \frac{1}{1+i} \), we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 1+i \) is \( 1-i \). Hence, \( \frac{1}{1+i} \times \frac{1-i}{1-i} = \frac{1(1-i)}{(1+i)(1-i)} = \frac{1-i}{1^2 - i^2} = \frac{1-i}{1 - (-1)} = \frac{1-i}{2} \).

Step 2: Simplify the Second Term

Now simplify \( \frac{1}{1-i} \) by multiplying the numerator and the denominator by the conjugate of the denominator, which is \( 1+i \). Thus, \( \frac{1}{1-i} \times \frac{1+i}{1+i} = \frac{1(1+i)}{(1-i)(1+i)} = \frac{1+i}{1^2 - i^2} = \frac{1+i}{1 - (-1)} = \frac{1+i}{2} \).

Step 3: Subtract the Two Expressions

Now we subtract the simplified expressions: \( \frac{1-i}{2} - \frac{1+i}{2} = \frac{1-i - (1+i)}{2} \). Simplifying the numerator gives us \( 1-i - 1 - i = -2i \). So, the expression becomes \( \frac{-2i}{2} = -i \).

Step 4: Write the Result in the Form \(a + b i\)

Finally, express your result \( -i \) in the form \( a + bi \). Here, \( a = 0 \) and \( b = -1 \). Thus, the expression is \( 0 + (-1)i = 0 - i \).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \( i \), is a fundamental concept in understanding complex numbers. It is defined by the equation \( i^2 = -1 \). This might seem a bit abstract initially because we're accustomed to thinking of squares as non-negative numbers. However, \( i \) allows us to extend the number system to solve problems that would otherwise be impossible with just real numbers.

• The imaginary unit \( i \) essentially acts as the "unit identity" for imaginary numbers, just as 1 does for real numbers.
• Every complex number has a real part and an imaginary part, typically expressed as \( a + bi \), where \( a \) and \( b \) are real numbers.

Understanding \( i \) enriches mathematics because it helps to handle roots of negative numbers, solve polynomial equations, and work with oscillating signals in fields like engineering and physics.

Conjugate

A core understanding in complex numbers is the concept of conjugates. A conjugate is obtained by changing the sign of the imaginary part of a complex number. For example, the conjugate of \( 1 + i \) is \( 1 - i \). This property is very useful when it comes to division and rationalizing denominators in expressions involving complex numbers.

• Multiplying a complex number by its conjugate results in a real number.
• This conjugate product equals the sum of the squares of the real and imaginary parts, expressed as \( a^2 + b^2 \).
• Using conjugates arises in simplifying fractional expressions in complex arithmetic.

In our exercise, conjugates allow us to clear the imaginary terms from the denominator, essentially "rationalizing" the expression to make it easier to work with.

Rationalizing the Denominator

Rationalizing the denominator is an important technique in mathematics that involves removing the imaginary components from a denominator. This is achieved by multiplying the numerator and denominator by the conjugate of the denominator.

• The goal here is to leave the denominator as a real number.
• In the example \( \frac{1}{1+i} \), multiplying by the conjugate \( 1-i \) simplifies the denominator to \( 1^2 - i^2 = 2 \).

By doing so, we convert complex expressions into simpler forms, which are easier to interpret, manage, or integrate into broader calculations. After simplification, we use basic arithmetic operations to process the remaining terms.

Complex Subtraction

Complex subtraction involves subtracting one complex number from another. This requires the subtraction of both the real components and imaginary components separately. Given the problem of subtracting two simplified complex numbers\( \frac{1-i}{2} - \frac{1+i}{2} \):

• Subtract each corresponding component: \( (1 - 1) - (i + i) = 0 - 2i \).
• Write the result as a simplified complex number: \( -i \).

The operation demonstrates the importance of handling real and imaginary parts independently. It's straightforward yet crucial to ensuring accurate solutions in complex arithmetic expressions. This separation is key to not only subtraction but to all arithmetic operations involving complex numbers.