Problem 61 - Algebra A Combined Function

Question

Divide. Write your answers in the form \(a+b i\) $$ \frac{3+5 i}{1+i} $$

Step-by-Step Solution

Answer

The answer is \(4+i\).

Step 1: Identify the Conjugate

First, identify the conjugate of the denominator. The original denominator is \(1+i\), so its conjugate is \(1-i\).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator and the denominator by the conjugate of the denominator. This gives us: \[\frac{(3+5i)(1-i)}{(1+i)(1-i)}\]

Step 3: Expand the Numerator

Distribute in the numerator: \[(3+5i)(1-i) = 3 \cdot 1 + 3 \cdot (-i) + 5i \cdot 1 + 5i \cdot (-i)\]Simplifying, we get: \[3 - 3i + 5i - 5i^2\]Since \(i^2 = -1\), this turns into: \[3 - 3i + 5i + 5 = 8 + 2i\]

Step 4: Simplify the Denominator

Expand the denominator: \[(1+i)(1-i) = 1 \cdot 1 + 1 \cdot (-i) + i \cdot 1 + i \cdot (-i)\]Simplify the expression: \[1 - i + i - i^2\]Since \(i^2 = -1\), it simplifies further to: \[1 + 1 = 2\]

Step 5: Write the Final Answer

Divide both the real and imaginary components of the numerator by the denominator: \[\frac{8 + 2i}{2} = \frac{8}{2} + \frac{2i}{2} = 4 + i\]Thus, the solution in the form \(a+bi\) is \(4+i\).

Key Concepts

Division of Complex Numbers

When dividing complex numbers, it's all about transforming a complex division problem into a simpler form. The strategy involves eliminating the imaginary unit from the denominator.
To do this, we use the conjugate of the denominator. The process typically consists of these steps:

• Identify the conjugate: Find the conjugate of the denominator. If the denominator is in the form of \(a+bi\), its conjugate is \(a-bi\).
• Multiply: Multiply both numerator and denominator by this conjugate.
• Simplify: Simplify the resulting expressions.

For example, in the division problem \(\frac{3+5i}{1+i}\), the goal is to remove the imaginary part in the denominator so the division can be more straightforward. Multiplying the numerator and denominator by the conjugate \(1-i\) allows us to accomplish this. This changes the division into a simpler fraction with a real denominator. The final solution can then be expressed in standard complex form \(a+bi\).
This method helps keep complex number division clear and structured.

Conjugate

A conjugate of a complex number is quite straightforward. Given any complex number \(a+bi\), its conjugate is \(a-bi\).
The conjugate switches the sign of the imaginary part while keeping the real part the same.
Conjugates are important in complex number division because:

• They help in rationalizing denominators of complex fractions.
• Multiplying a complex number by its conjugate always results in a real number.

For instance, if you have the complex number \(1+i\), its conjugate would be \(1-i\).
By multiplying the numerator and denominator by the conjugate, you effectively remove the imaginary part from the denominator, making the division process mathematically simpler.Conjugates are a powerful tool to transform an initially complicated problem into a much easier one.

Imaginary Unit

The imaginary unit, symbolized by \(i\), is a unique concept in mathematics with a simple definition yet profound implications.
It is defined by the property \(i^2 = -1\).
This property differentiates it from real numbers and is central to the arithmetic of complex numbers.

• When dealing with complex arithmetic, the power of \(i\) loops in a predictable cycle: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).
• This cyclical behavior makes simplifying expressions with imaginary units crucial in complex number operations.

In the division of complex numbers, simplifying terms using \(i^2 = -1\) is essential—a prerequisite to deriving final expressions like \(a+bi\).
Understanding and working with the imaginary unit underpins successful manipulation and resolution of complex number problems, offering insights into solutions that are not apparent within the realm of real numbers alone.