Problem 28 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The result is \(\frac{1}{2} - \frac{1}{2}i\).

Step 1: Identify the Conjugate

To simplify the given expression \(\frac{1}{1+i}\), we need to eliminate the imaginary part from the denominator by multiplying both numerator and denominator by the conjugate of \(1+i\). The conjugate of \(1+i\) is \(1-i\).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate: \(\frac{1}{1+i} \times \frac{1-i}{1-i} = \frac{1(1-i)}{(1+i)(1-i)}\).

Step 3: Simplify the Denominator

Expand the denominator using the difference of squares formula: \((1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2\). So, the expression becomes \(\frac{1-i}{2}\).

Step 4: Separate into Real and Imaginary Parts

Divide the expression \(\frac{1-i}{2}\) into its real and imaginary components: \(\frac{1}{2} - \frac{1}{2}i\).

Step 5: Write the Result in the Form \(a+bi\)

The expression \(\frac{1}{2} - \frac{1}{2}i\) can be written in the required form \(a+bi\) where \(a = \frac{1}{2}\) and \(b = -\frac{1}{2}\).

Key Concepts

Conjugate

Understanding the concept of the conjugate of a complex number is essential in simplifying complex expressions, particularly when dealing with division. A complex number is typically in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The conjugate of this number would be \(a - bi\). Simply put, the conjugate reflects the imaginary part of a complex number across the real axis by changing the sign of the imaginary part.

For example, if you have a complex number like \(1 + i\), its conjugate would be \(1 - i\). When we multiply a complex number by its conjugate, we eliminate the imaginary part because it results in a real number. This is particularly useful in division as it rationalizes the denominator, which means we express the division without an imaginary number in the denominator.

• Example: Conjugate of \(3 + 4i\) is \(3 - 4i\).
• Multiplying a complex number by its conjugate: \((1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 2\).

Using conjugates is a systematic approach to ensure clarity in expressions and avoid complexities where imaginations are involved in the denominator.

Real and Imaginary Parts

Every complex number has two main components: its real part and its imaginary part. These components are crucial for expressing complex numbers in standard form and are helpful when performing mathematical operations like addition, subtraction, and division. A complex number is represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

When dealing with operations involving complex numbers, it's important to break them down into their real and imaginary components. For instance, in expressions resulting from division, like \(\frac{1-i}{2}\), this can be broken down into its real and imaginary parts as follows:

• The real part is \(\frac{1}{2}\)
• The imaginary part is \(-\frac{1}{2}i\)

By expressing the number in this form \(a + bi\), it clarifies the magnitude of the real and imaginary components, facilitating easier analysis and usage in further calculations. Understanding these parts helps in graphically representing complex numbers on the complex plane, where the real part corresponds to the x-axis and the imaginary part to the y-axis.

Simplification of Expressions

Simplifying expressions involving complex numbers often requires converting them into a more manageable form, such as \(a + bi\). This simplification process not only makes the expression easier to understand but also simplifies subsequent mathematical operations like addition, subtraction, and multiplication.

Let's consider a division problem like \(\frac{1}{1+i}\). Directly handling the division with an imaginary denominator can be cumbersome, so we use the conjugate to rationalize the denominator. After multiplying both numerator and denominator by the conjugate \(1-i\), the problem turns into \(\frac{1-i}{2}\).

• The numerator \(1-i\) remains in its complex form, while
• the denominator \(2\), a real number, results from multiplying \((1+i)(1-i)\).

Further simplification results in separating the terms into the real part \(\frac{1}{2}\) and the imaginary part \(-\frac{1}{2} i\), representing the complex number in the form \(a+bi\). This process of simplification via conjugates and separation clarifies the expression and is an essential technique when dealing with complex numbers.