Problem 73 Find each quotient. $$ \frac... [FREE SOLUTION] (2024)

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Chapter 8: Problem 73

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

Short Answer

Expert verified

$1 + 3i$

Step by step solution

Understand the problem

The task is to find the quotient of the complex numbers $\frac{3-i}{-i}$. This involves division of complex numbers.

Multiply by the conjugate of the denominator

To simplify the division of complex numbers, multiply both numerator and denominator by the conjugate of the denominator. The conjugate of $-i$ is $i$. Thus, multiply both numerator and denominator by $i$: \[\frac{(3-i) \times i}{-i \times i}\]

Simplify the denominator

Calculate the denominator: \[-i \times i = -i^2 = -(-1) = 1\]

Distribute in the numerator

Multiply out the terms in the numerator: \[(3-i) \times i = 3i - i^2\]

Simplify the numerator

Recall that $i^2 = -1$, so substitute this value in: \[3i - (-1) = 3i + 1\]

Divide by the simplified denominator

Now divide the simplified numerator by the simplified denominator: \[\frac{3i + 1}{1} = 3i + 1\]

Key Concepts

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

complex numbers

A complex number is a number that has both a real part and an imaginary part. It is usually written in the form:
\[ z = a + bi \]
where a and b are real numbers, and i is the imaginary unit, which we'll discuss more soon.
The real part of the complex number is a, and the imaginary part is the term that includes i, which is bi.
Let's look at an example:

If a is 3 and b is -4, the complex number is 3 - 4i.

Complex numbers can be used in various fields, such as engineering and physics, to represent phenomena that cannot be explained by real numbers alone.

conjugate

The conjugate of a complex number is another complex number with its imaginary part's sign reversed.
Given a complex number \[ z = a + bi \], its conjugate is written as \[ \overline{z} = a - bi \].
Conjugates are useful in various calculations, including the division of complex numbers, as shown in the exercise step where i was used to multiply the numerator and the denominator.
Another important use of conjugates is to remove the imaginary part from the denominator of a fraction, which simplifies division. For instance, if you want to divide by z = a + bi, you can use its conjugate to form a real denominator.

imaginary unit

The imaginary unit is a mathematical concept denoted by the symbol i.
It is defined as the square root of -1: \[ i = \sqrt{-1} \].
Because no real number squared equals a negative number, i is called an 'imaginary' number.
When squared, \[ i^2 = -1 \].
Complex numbers utilize the imaginary unit to extend the real numbers, allowing for a more comprehensive understanding of mathematics and its applications.
For example, in the exercise above, the imaginary unit i was used in the numerator and denominator to simplify the division process. The property \[ i^2 = -1 \] was crucial in simplifying the expression.

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$Problem 73 Find each quotient. $$ \frac... [FREE SOLUTION] (3)$

Most popular questions from this chapter

A Sanyo color television, model AVM-2755, has a rec tangular screen with a21.7 -in. width. Its height is 16 in. What is the measure of the diagonal ofthe screen, to the nearest tenth of an inch?Solve each equation. $$ -6 x^{2}+7 x=-10 $$Perform the indicated operations. Give answers in standard form. $$ \frac{2}{3+4 i}+\frac{4}{1-i} $$If the sides of a triangle are $\sqrt{65}$ in. $, \sqrt{35}$ in., and$\sqrt{26}$ in., which one of the following is the best estimate of itsperimeter? A. 20 in. B. 26 in. C. 19 in. D. 24 in.Simplify each radical. $\frac{2}{4+\sqrt{3}}$

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