Problem 36 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form \(a+b i .\) $$ \frac{-3+5 i}{15 i} $$

Step-by-Step Solution

Answer

The result is \(-\frac{1}{3} + \frac{1}{5}i\).

Step 1: Multiply the Numerator and Denominator by the Conjugate of the Denominator

The current expression is \( \frac{-3+5i}{15i} \). Multiplying both the numerator and the denominator by \(-i\) (the conjugate of \(15i\)) helps remove the imaginary component from the denominator. Thus, we will perform the following operation:\(\frac{(-3+5i)(-i)}{15i(-i)} = \frac{3i-5}{-15}\).

Step 2: Simplify the Denominator

The denominator now is \(-15\), since \(15i(-i) = 15(-i^2) = 15(1) = 15\). So, the expression simplifies to:\(\frac{-3i + 5}{15}\).

Step 3: Simplify Each Part of the Result

Separate the numerator and divide each term by \(-15\):- Real part: \(\frac{5}{-15} = -\frac{1}{3}\)- Imaginary part: \(\frac{-3i}{-15} = \frac{1}{5}i\)Thus, the expression becomes \(-\frac{1}{3} + \frac{1}{5}i\).

Key Concepts

Understanding the Imaginary Unit

When exploring complex numbers, the imaginary unit is a key player. Denoted by the letter \( i \), the imaginary unit is defined with the property that \( i^2 = -1 \).
This is crucial because it allows the extension of the number system beyond real numbers, enabling solutions to equations that don’t have real solutions, such as \( x^2 + 1 = 0 \).

Here's how it works in practice:

• Imaginary numbers are essentially real numbers multiplied by the imaginary unit \( i \). For example, \(5i\) means 5 times the imaginary unit.
• Imaginary numbers combine with real numbers to form complex numbers, which can be expressed as \(a + bi\), where \(a\) and \(b\) are real numbers.

It’s important to grasp this concept as it lays the groundwork for understanding complex arithmetic and operations, like those seen in the given exercise.

Exploring Complex Conjugates

A complex conjugate is a straightforward yet powerful tool in complex number calculations.
For a given complex number \(a + bi\), its complex conjugate is \(a - bi\).

The role of a complex conjugate is pivotal for eliminating the imaginary parts from denominators when dividing complex numbers. Using a complex conjugate, we achieve a real number in the denominator.
The product of a complex number and its conjugate is always a real number, specifically:

• The sum of the squares of its real and imaginary parts, formulated as \((a+bi)(a-bi) = a^2 + b^2\).

Applying this concept was crucial in the exercise. By multiplying the numerator and denominator of the expression by \(-i\) (the conjugate of \(15i\)), the denominator becomes a real number, allowing for straightforward simplification.

Rationalizing Denominators

Rationalizing the denominator involves a technique used to clear complex numbers or radicals from the denominator of a fraction.
For complex numbers, this typically involves using the concept of the complex conjugate.

The aim is to achieve a real number in the denominator, making the fraction more manageable and standard.
This process involves:

• Identifying the complex number or radical in the denominator.
• Multiplying both the numerator and the denominator by the conjugate of the denominator.

In the exercise, we multiplied both the numerator and the denominator by \(-i\), eliminating the imaginary unit from the denominator. This made further simplification possible, resulting in the expression \(-\frac{1}{3} + \frac{1}{5}i\).
Rationalizing helps present complex numbers in a more conventional form, improving comprehension and manipulation.