Problem 44 - Algebra 2 and Trigonometry

Question

In \(38-45,\) find the multiplicative inverse of each of the following in \(a+b i\) form. $$ \frac{5}{6}+3 i $$

Step-by-Step Solution

Answer

The multiplicative inverse is \( \frac{30}{349} - \frac{108}{349}i \).

Step 1: Understand the formula for the multiplicative inverse

The multiplicative inverse of a complex number \(a + bi\) is given by \(\frac{a}{a^2 + b^2} - \frac{b}{a^2 + b^2}i\). This formula involves multiplying the number by its conjugate and dividing by the magnitude squared.

Step 2: Identify real and imaginary parts

For the complex number \(\frac{5}{6} + 3i\), identify the real part \(a\) as \(\frac{5}{6}\) and the imaginary part \(b\) as \(3\).

Step 3: Calculate the magnitude squared

The magnitude squared \(a^2 + b^2\) is calculated as: \[\left(\frac{5}{6}\right)^2 + (3)^2 = \frac{25}{36} + 9 = \frac{25}{36} + \frac{324}{36} = \frac{349}{36}\]

Step 4: Use the formula to find the multiplicative inverse

The multiplicative inverse is given by:\[ \frac{\frac{5}{6}}{\frac{349}{36}} - \frac{3}{\frac{349}{36}}i \] Simplifying each term:\[ \frac{1}{\frac{349}{36}} = \frac{36}{349} \] Thus:\[ \frac{5}{6} \times \frac{36}{349} = \frac{30}{349} \]\[ 3 \times \frac{36}{349} = \frac{108}{349} \] Resulting in the conjugate:\[ \frac{30}{349} - \frac{108}{349}i \]

Step 5: Simplify the result

Combine the terms to arrive at the solution in standard form:\[ \frac{30}{349} - \frac{108}{349}i \] This is the multiplicative inverse of \( \frac{5}{6} + 3i \).

Key Concepts

Complex Numbers

Complex numbers are fascinating as they extend our number system beyond the familiar real numbers. A complex number is any number that can be written in the form of \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) represents the imaginary unit. The imaginary unit, \( i \), is defined by the property \( i^2 = -1 \). This property is what allows us to step into the realm of imaginary numbers, bridging a gap left by the inability to square root negative numbers in the set of real numbers.
A complex number has two distinct parts:

• The real part: This is denoted by \( a \).
• The imaginary part: This is denoted by \( bi \), where \( b \) is a real number.

These numbers are especially useful in various fields such as electrical engineering and fluid dynamics due to their ability to represent oscillations and waves.

Conjugate

The conjugate of a complex number is like its mirror image across the real axis on the complex plane. If you have a complex number \( a + bi \), its conjugate is \( a - bi \).
This simple operation of flipping the sign of the imaginary part helps a lot in complex arithmetic, especially when dividing complex numbers.

• When you multiply a complex number by its conjugate, the result is a real number.
• This process helps find the magnitude squared of a complex number, which is instrumental in computing its multiplicative inverse.

When working with the conjugate during this process, it often simplifies expressions and aids in obtaining real results.

Magnitude Squared

The magnitude of a complex number is like its length or distance from the origin in the complex plane. To find the magnitude squared of a complex number \( a + bi \), you use the formula \( a^2 + b^2 \).
For the complex number \( \frac{5}{6} + 3i \), the magnitude squared calculation follows these steps:

• Square the real part: \((\frac{5}{6})^2\).
• Square the imaginary part: \(3^2\).
• Add them together to get \( \frac{25}{36} + 9 = \frac{349}{36} \).

This result, the magnitude squared, is essential for dividing complex numbers and forms the basis of the denominator when calculating the multiplicative inverse.

Real and Imaginary Parts

The real and imaginary parts are fundamental to understanding complex numbers. Identifying these parts is crucial for performing operations like addition, subtraction, and multiplication.
For example, in the complex number \( \frac{5}{6} + 3i \):

• The real part \( a \) is \( \frac{5}{6} \).
• The imaginary part \( b \) is \( 3 \), associated with \( i \).

The distinction between these parts allows us to utilize specific rules for complex arithmetic. Multiplicative inverse, which relies heavily on both the real and imaginary components, showcases how these distinct parts work in harmony to achieve a result that renders the original number's product with its inverse to be 1.