Problem 48 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result is \( 0 - i \).

Step 1: Recognize Complex Numbers

Identify the complex numbers in the expression: \( 1 - \sqrt{-1} \) and \( 1 + \sqrt{-1} \). Here, \( \sqrt{-1} \) represents the imaginary unit \( i \). Thus, the expression is \( \frac{1 - i}{1 + i} \).

Step 2: Multiply by the Conjugate

To simplify the fraction, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 1 + i \) is \( 1 - i \). So, multiply both parts by \( 1 - i \):\[ \frac{(1-i)(1-i)}{(1+i)(1-i)} \]

Step 3: Simplify the Denominator

The denominator \((1+i)(1-i)\) is a difference of squares: \[ (1)^2 - (i)^2 = 1 - (-1) = 2 \]

Step 4: Simplify the Numerator

Expand the numerator \((1-i)(1-i)\):\[ 1 - 2i + i^2 \] Recall that \( i^2 = -1 \), so it becomes:\[ 1 - 2i - 1 = -2i \]

Step 5: Write the Simplified Expression

Combine the results:\[ \frac{-2i}{2} = -i \]

Step 6: Express in the form \(a + bi\)

Rewrite the simplified result \(-i\) in the form \( a + bi \), where \( a = 0 \) and \( b = -1 \). Thus, it's equivalent to \( 0 - i \).

Key Concepts

Conjugate of a Complex Number

In the realm of complex numbers, the conjugate is a key concept that helps simplify complex arithmetic. The conjugate of a complex number switches the sign of the imaginary part, making it very useful in rationalizing denominators in expressions. For any complex number expressed as \(a + bi\), its conjugate is \(a - bi\). This process aids in reducing expressions to simpler forms.

• The conjugate of \(1 + i\) is \(1 - i\).
• Multiplying a complex number by its conjugate results in a real number.
• Used mainly for rationalizing denominators in fractions containing complex numbers.

In our original exercise, we used the conjugate of \(1 + i\), which is \(1 - i\), to simplify the expression. Multiplying numerator and denominator by the conjugate ensures that the denominator becomes a real number, making it easier to simplify further.

Difference of Squares

The difference of squares is a powerful algebraic tool often used with polynomials and in simplifying complex numbers. The standard identity is \((a+b)(a-b) = a^2 - b^2\). When dealing with complex numbers, this identity helps in simplifying expressions by converting complex products into simpler real numbers.

• Essentially involves two terms where one is subtracted from the other, squared.
• In complex arithmetic, helps convert expressions with imaginary units into real numbers.
• The expression \((1+i)(1-i)\) is an example of a difference of squares, yielding \(1^2 - (i)^2\).

Remember that \(i^2 = -1\). By applying the difference of squares to the denominator \((1+i)(1-i)\) in this exercise, we transformed it into the real number 2, facilitating further simplification.

Imaginary Unit

The imaginary unit, denoted as \(i\), is the cornerstone of complex numbers. It is defined by its property \(i^2 = -1\), which sets the foundation for constructing all other complex numbers. Every complex number consists of a real part and an imaginary part.

• Imaginary numbers are expressed as multiples of \(i\), for example, \(-2i\).
• Adding and subtracting imaginary numbers follows similar rules to real numbers.
• Simplifies polynomial equations that have no real solutions by providing a way to include square roots of negative numbers.

In the example given in this exercise, \(\sqrt{-1}\) is recognized as \(i\). This insight allows one to treat expressions involving square roots of negative numbers within the framework of complex arithmetic, enabling further algebraic manipulation and simplification, as shown when rewriting \( -\sqrt{-1} \) as \(-i\).