Problem 39 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ \frac{1}{1+i} $$

Step-by-Step Solution

Answer

The expression in standard form is \( \frac{1}{2} - \frac{1}{2}i \).

Step 1: Identify the Problem

We need to write the expression \( \frac{1}{1+i} \) in standard form, which is \( a + bi \) where \( a \) and \( b \) are real numbers.

Step 2: Multiply by the Conjugate

To eliminate the imaginary part from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, which is \( 1-i \). This gives us: \[\frac{1}{1+i} \times \frac{1-i}{1-i} = \frac{1-i}{(1+i)(1-i)}\]

Step 3: Simplify the Denominator

Multiply the denominator using the difference of squares formula: \[(1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2.\]Thus, the denominator simplifies to 2.

Step 4: Simplify the Numerator

Now multiply the expression:\( 1-i \), resulting in \( 1 - i \). As there are no like terms to combine, it remains \( 1 - i\).

Step 5: Combine Everything

Combine the simplified numerator and denominator:\[\frac{1-i}{2} = \frac{1}{2} - \frac{i}{2}.\]Written in standard form, the expression is \( \frac{1}{2} - \frac{1}{2}i \).

Key Concepts

Standard Form

In the realm of complex numbers, writing a number in standard form is essential for clarity and consistency. The standard form of a complex number is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). This form helps in identifying both parts of a complex number easily.

• \( a \) is known as the real part.
• \( bi \) is the imaginary part.

For instance, in the expression \( \frac{1}{2} - \frac{1}{2}i \), we have \( a = \frac{1}{2} \) and \( b = -\frac{1}{2} \). This means the real part is \( \frac{1}{2} \) and the imaginary part is \( -\frac{1}{2}i \). Hence, the complex number is clearly displayed in its standard form.

Conjugate

The concept of the conjugate is an essential tool when working with complex numbers, especially for simplifying expressions. For any complex number \( a + bi \), its conjugate is \( a - bi \). The conjugate essentially flips the sign of the imaginary part.

• Given the complex number \( 1+i \), its conjugate is \( 1-i \).
• Multiplying a complex number by its conjugate removes the imaginary part, simplifying calculations.

This property is particularly useful when you need to rationalize denominators, as seen in the original exercise. By multiplying \( 1+i \) by its conjugate \( 1-i \), the expression simplifies significantly, making it easier to handle.

Imaginary Numbers

Imaginary numbers come into play when dealing with complex numbers. They are built on the imaginary unit \( i \), which satisfies the equation \( i^2 = -1 \). Imaginary numbers are of the form \( bi \), where \( b \) is a real number.

• The key property is that an imaginary number squared is always a negative real number.
• For example, \( i^2 = -1 \); \( 2i \cdot 2i = -4 \).

Imaginary numbers, when combined with real numbers, form complex numbers. They are incredibly useful in a wide array of mathematical applications, including engineering and physics, among others.

Difference of Squares

The difference of squares is a handy algebraic identity that simplifies the multiplication of expressions in the form \((a + b)(a - b)\). It states:\[(a + b)(a - b) = a^2 - b^2\]This identity is particularly useful when working with complex numbers, especially when dealing with conjugates. In the exercise, it was used to simplify the denominator:

• \((1+i)(1-i) = 1^2 - (i^2)\)
• This simplifies to \(1 - (-1) = 2\).

By applying the difference of squares, we quickly eliminate the imaginary components in the denominator, allowing the expression to be written in a simpler standard form.