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Chapter 10: Problem 73

Find each quotient. $$ \frac{3-i}{-i} $$

Short Answer

Expert verified
3i + 1

Step by step solution

01

- Understand the Problem

The task is to find the quotient of the complex numbers \( \frac{3-i}{-i} \). This involves dividing the numerator by the denominator.
02

- Multiply by Conjugate of the Denominator

Multiply both numerator and denominator by the conjugate of the denominator. The conjugate of \( -i \) is \( i \). Thus, multiply by \( \frac{i}{i} \): \[ \frac{3-i}{-i} \times \frac{i}{i} \]
03

- Distribute and Simplify

Distribute \( i \) over the numerator \( (3-i) \) and calculate the denominator: \[ \frac{(3-i) \times i}{-i \times i} \]\[ (3 \times i) - (i \times i) = 3i - i^2 \]Since \( i^2 = -1 \): \[ 3i - (-1) = 3i + 1 \]The denominator simplifies to \[ -i \times i = -i^2 = 1 \]
04

- Write the Final Answer

Now we have: \[ \frac{3i + 1}{1} = 3i + 1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complex Number Operations
Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form:
Conjugate of Complex Numbers
To better understand complex numbers, we need to know about their conjugates. The conjugate of a complex number changes the sign of the imaginary part. For example, for a complex number of the form:
Simplifying Complex Fractions
Once you understand how to handle complex number operations and their conjugates, the next step is simplifying complex fractions. When you divide one complex number by another, you often get a complex fraction. Here are the steps to simplify it:

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