Problem 41 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ \frac{4+i}{5-i} $$

Step-by-Step Solution

Answer

The expression in standard form is \(\frac{19}{26} + \frac{9}{26}i\).

Step 1: Identify the Conjugate

To remove the imaginary part from the denominator, multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(5-i\) is \(5+i\).

Step 2: Multiply Numerator and Denominator by the Conjugate

Multiply both the numerator \(4+i\) and the denominator \(5-i\) by \(5+i\):\[\frac{(4+i)(5+i)}{(5-i)(5+i)}\]

Step 3: Simplify the Denominator

Apply the formula \((a-b)(a+b) = a^2 - b^2\) to simplify the denominator:\[(5-i)(5+i) = 5^2 - i^2 = 25 + 1 = 26\]

Step 4: Expand and Simplify the Numerator

Use the distributive property to expand \((4+i)(5+i)\):\[(4)(5) + (4)(i) + (i)(5) + (i)(i) = 20 + 4i + 5i + i^2\]Since \(i^2 = -1\), the expression becomes:\[20 + 9i - 1 = 19 + 9i\]

Step 5: Write the Expression in Standard Form

Combine your simplified steps:\[\frac{19 + 9i}{26}\]This can be written in standard form as \(\frac{19}{26} + \frac{9}{26}i\).

Key Concepts

Standard Form

When dealing with complex numbers, it's essential to understand how they are expressed in standard form. The standard form of a complex number is given by \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This format makes it clear that a complex number consists of both a real part and an imaginary part.

• Real Part: In the expression \(a + bi\), \(a\) is called the real part.
• Imaginary Part: The term \(bi\) is called the imaginary part. Here, \(b\) is a real coefficient.

Understanding standard form helps in a variety of mathematical operations, such as addition, subtraction, and especially in simplifying expressions involving complex numbers.
In the given exercise, we took the expression \(\frac{19}{26} + \frac{9}{26}i\), which is in standard form since it clearly separates the real part \(\frac{19}{26}\) from the imaginary part \(\frac{9}{26}i\). This allows for easier computation and understanding of the complex number.

Imaginary Unit

The concept of the imaginary unit is central to working with complex numbers. The imaginary unit is denoted by \(i\). It is defined by the property \(i^2 = -1\). This is a crucial component when performing operations involving complex numbers because it fundamentally changes how multiplication and other operations are conducted.
Some quick facts about the imaginary unit:

• Expression: The term \(bi\) in a complex number depicts how many times the imaginary unit is included, scaled by a real number \(b\).
• Square of \(i\): Since \(i^2 = -1\), squaring imaginary numbers affects their phase, often bringing them back into the negative real axis.

In our exercise, identifying \(i^2\) as \(-1\) was essential when simplifying the product \((4+i)(5+i)\), leading to the transformation of terms and simplifying the expression into a standard form.

Conjugate

A conjugate is a concept used to simplify the division of complex numbers, especially when aiming to remove imaginary components from the denominator. For any complex number \(a + bi\), its conjugate will be \(a - bi\). Conjugates are helpful because multiplying a complex number by its conjugate results in a real number.

• Real Part Remains: When forming a conjugate, the real part remains unchanged.
• Imaginary Part Changes Sign: The sign of the imaginary part is reversed.

In the exercise, the conjugate of the denominator \(5 - i\) was \(5 + i\). By multiplying the top and bottom of the fraction by this conjugate, the imaginary part in the denominator is eliminated through the difference of squares formula \((a-b)(a+b) = a^2 - b^2\). This simplification process was critical to rewriting the overall expression in standard form.