Problem 61 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the quotient in standard form. $$\frac{i}{(4-5 i)^{2}}$$

Step-by-Step Solution

Answer

The quotient in standard form is \(\frac{5}{41}+\frac{4}{41}i\).

Step 1: Identify the Conjugate

We start by identifying the conjugate of the denominator. The conjugate of a complex number a+bi is a-bi. Therefore, the conjugate of 4-5i is 4+5i.

Step 2: Multiply by the Conjugate

We then multiply both the numerator and denominator by the conjugate of the denominator. This gives us: \(\frac{i(4+5i)}{(4-5i)(4+5i)}\).

Step 3: Simplify the Expression

After multiplication, we simplify the expression to get a standard form of the complex number. The multiplication of (4-5i)(4+5i) gives 41. Applying foil method to the numerator yields 4i + 5. Simplifying gives us: \(\frac{4i + 5}{41}\).

Step 4: Write in standard form a + bi

To write the quotient in standard form a + bi, parse the real and imaginary parts. This allows us to rewrite the expression as: \(\frac{5}{41}+\frac{4}{41}i\)

Key Concepts

Standard Form

Standard form for complex numbers refers to the format expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This form makes it easy to perform operations on complex numbers.
Here's why it's useful:

• Clarity: Having a structured form helps to clearly identify the real and imaginary components.
• Operations: It simplifies addition, subtraction, and comparison of complex numbers.
• Graphing: Standard form makes it easier to plot on the complex plane, where the real part is the x-coordinate and the imaginary part is the y-coordinate.

In the given exercise, after performing the arithmetic operation, the complex number is expressed in standard form as \(\frac{5}{41} + \frac{4}{41}i\). Here, \(\frac{5}{41}\) represents the real part while \(\frac{4}{41}\) multiplied by \(i\) gives the imaginary part.
Always aim to have your final complex number in this format for simplicity and consistency.

Complex Conjugate

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). This simple transformation plays a crucial role in operations involving complex numbers.

• Purpose: It is used to eliminate the imaginary part of a denominator in a fraction, thereby simplifying division involving complex numbers.
• Property: Multiplying a complex number by its conjugate results in a real number. For example, \((4-5i)(4+5i) = 41\).

In our exercise, we used the complex conjugate \(4 + 5i\) to multiply the denominator \(4 - 5i\), resulting in a purely real number, 41. This step is essential to clearing the imaginary unit \(i\) from the denominator, which is necessary when simplifying any complex fraction.

FOIL Method

The FOIL method is a technique used primarily to expand two binomials. It stands for First, Outer, Inner, Last, representing the pairs of terms multiplied together.
For a product of two binomials, such as \((x + y)(a + b)\), apply the FOIL steps:

• First: Multiply the first terms from each binomial together, \(x \cdot a\).
• Outer: Multiply the two outer terms, \(x \cdot b\).
• Inner: Multiply the two inner terms, \(y \cdot a\).
• Last: Multiply the last terms from each binomial, \(y \cdot b\).

Add these results to expand the expression.
In our problem, the numerator \(i(4 + 5i)\) was expanded using the FOIL method:

• First: \(i \cdot 4 = 4i\)
• Outer: \(i \cdot 5i = 5i^2 = -5\) (since \(i^2 = -1\))
• Inner and Last: These do not apply here directly, as one term is nullified by previous multiplication.

This results in \(4i - 5\), which is later incorporated into the standard form by separating the real and imaginary components.