Problem 45 - Algebra 2 and Trigonometry

Question

In \(38-45,\) find the multiplicative inverse of each of the following in \(a+b i\) form. $$ 9+\pi i $$

Step-by-Step Solution

Answer

The inverse is \(\frac{9}{81 + \pi^2} - \frac{\pi}{81 + \pi^2}i\).

Step 1: Identify the Complex Number Parts

The given complex number is \(9 + \pi i\), where the real part \(a = 9\) and the imaginary part \(b = \pi\).

Step 2: Define the Multiplicative Inverse

The multiplicative inverse of a complex number \(a + bi\) is given by \(\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i\).

Step 3: Calculate \(a^2 + b^2\)

Compute the value of \(a^2 + b^2\):\[a^2 + b^2 = 9^2 + (\pi)^2 = 81 + \pi^2\]

Step 4: Apply to the Inverse Formula

Substitute \(a = 9\) and \(b = \pi\) in the inverse formula:\[\text{Inverse} = \frac{9}{81 + \pi^2} - \frac{\pi}{81 + \pi^2}i\]

Step 5: Simplify the Result

Express the result concisely as:\[\frac{9}{81 + \pi^2} \quad - \quad \, \frac{\pi}{81 + \pi^2}i\]

Key Concepts

Multiplicative Inverse

The multiplicative inverse of a complex number is an intriguing concept that breaks down quite neatly with the right approach.
When dealing with complex numbers, say in the form of \(a + bi\), the multiplicative inverse is defined as the number that, when multiplied by \(a + bi\), results in 1 (the identity element for multiplication).
This inverse is found using the formula: \[\text{Inverse} = \frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i\]Understanding this formula is crucial:

• Each part of the inverse is the original part divided by the sum \(a^2 + b^2\).
• This effectively involves both parts of the complex number, the real \(a\) and the imaginary \(b\), making sure each has its place in the result.

Practically, this ensures that multiplying \((a + bi)\) by its inverse yields 1.
Unlike real numbers, where the inverse is simply \(1/a\), the complex multiplicative inverse accounts for the two-dimensional nature of complex numbers.

Real and Imaginary Parts

To navigate the world of complex numbers, distinguishing between the real and imaginary components is essential.
A complex number is typically denoted as \(a + bi\), where:

• \(a\) represents the real part.
• \(bi\) signifies the imaginary part, with \(b\) being real and \(i\) denoting the square root of -1.

In our example, \(9 + \pi i\):

• The real part is 9.
• The imaginary part is \(\pi\fai\) (or simply \pi when focusing on the coefficient).

Recognizing these components allows for mathematical operations like addition, subtraction, multiplication, and finding inverses to proceed smoothly.
Grasping this separation is crucial, as mixing these components leads to misunderstandings in more complex calculations.

Complex Conjugate

The complex conjugate is a helpful tool in the realm of complex numbers, often providing a pathway to simplifying complex-number arithmetic.
The complex conjugate of a number \(a + bi\) is expressed as \(a - bi\).
Here's why it's important:

• It allows for the simplification of division involving complex numbers.
• When pairing a complex number with its conjugate, the imaginary parts disappear, leaving a purely real number.

This process results because:\[(a + bi)(a - bi) = a^2 + b^2\]This turns multiplication into a sum of squares without any imaginary components.
Complex conjugates are a key part of finding the multiplicative inverse, particularly in avoiding imaginary parts in denominators, which can complicate numerical computations further down the line.

a + bi Form

Understanding the \(a + bi\) form is fundamental in working with complex numbers.
This denotes a standard way of expressing any complex number, consisting of:

• \(a\): the real number component.
• \(bi\): the imaginary component, where \(i\) stands for the imaginary unit.

This form is pivotal for easy addition, subtraction, multiplication, and division of complex numbers because it clearly distinguishes between real and imaginary parts.
For example, in \(9 + \pi i\):

• \(9\) is the real part.
• \(\pi i\) contains the imaginary part.

Utilizing the \(a + bi\) form simplifies both the visualization and computation of complex numbers by organizing them plainly.
This structure also helps in transformation operations, like finding inverses, because each part can be operated on independently, maintaining clarity and accuracy throughout the process.