Problem 54 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The expression simplifies to \(-i\), or \(0 - 1i\).

Step 1: Identify the Components

Recognize that the expression involves complex numbers. The term \( \sqrt{-1} \) is equivalent to \( i \), the imaginary unit, where \( i^2 = -1 \). Rewrite the expression as \( \frac{1-i}{1+i} \).

Step 2: Rationalize the Denominator

To eliminate the imaginary unit \( i \) from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, \( 1 - i \). This gives: \( \frac{(1-i)(1-i)}{(1+i)(1-i)} \).

Step 3: Simplify Numerator and Denominator

Expand both the numerator and the denominator. The numerator becomes \((1-i)^2 = 1 - 2i + i^2 = 1 - 2i - 1 = -2i\). The denominator becomes \((1+i)(1-i) = 1 - i^2 = 1 + 1 = 2\).

Step 4: Simplify the Expression

Now, divide the simplified numerator by the denominator: \( \frac{-2i}{2} = -i \).

Step 5: Express in the Form a + bi

The result found is \(-i\), which can be written in the form \(0 - 1i\). Therefore, \( a = 0 \) and \( b = -1 \).

Key Concepts

Imaginary Unit

The imaginary unit, represented as \( i \), is a fundamental concept in complex numbers. It is defined by the equation \( i^2 = -1 \). This property is what distinguishes complex numbers from real numbers, allowing us to handle the square root of negative numbers. It’s helpful to think of the imaginary unit as a building block for constructing complex numbers.

• Complex numbers take the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.
• The imaginary unit doesn’t behave like real numbers due to its unique property \( i^2 = -1 \).

Whenever you encounter \( \sqrt{-1} \) in mathematical expressions, it can be replaced with \( i \). This was the first step recognized in our exercise, transforming the expression \( \frac{1 - \sqrt{-1}}{1 + \sqrt{-1}} \) into \( \frac{1-i}{1+i} \). This conversion is crucial for manipulation and simplification of expressions involving square roots of negative values.

Rationalizing the Denominator

Rationalizing the denominator is a technique used to eliminate the imaginary unit from the denominator of a fraction. This is particularly useful because having an imaginary number in the denominator can complicate further mathematical calculations.

• To rationalize a denominator containing \( i \), multiply both the numerator and the denominator by the complex conjugate of the denominator.
• The complex conjugate of a complex number \( a + bi \) is \( a - bi \). Using this, you can eliminate the imaginary part effectively.

With our example, we multiplied both the numerator and the denominator by \( 1 - i \), the conjugate of \( 1 + i \). This transforms the expression to \( \frac{(1-i)(1-i)}{(1+i)(1-i)} \), simplifying the denominator to a real number. This step ensures our final result \(-i\) is easy to interpret and use.

Complex Conjugate

The complex conjugate is an important concept when dealing with complex numbers. Given a complex number \( a + bi \), its conjugate is \( a - bi \). Conjugates have a special property that is incredibly useful: the product of a complex number and its conjugate is always a real number.

• The product \( (a + bi)(a - bi) = a^2 + b^2 \), which is purely real.
• This feature is leveraged when rationalizing denominators, as seen in our exercise.

By multiplying by the complex conjugate, the imaginary units cancel out in the denominator. In our given problem, multiplying the denominator \( 1+i \) by its conjugate \( 1-i \) results in \( 1 + 1 = 2 \), which notably lacks an \( i \) term. Remember, using the complex conjugate is often a go-to strategy to simplify expressions involving complex numbers.