Problem 77 - Precalculus Student Solutions Manual 5th

Question

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{8}{-i}$$

Step-by-Step Solution

Answer

The answer is \(0 + 8i\).

Step 1 - Eliminate the imaginary number from the denominator

To remove the imaginary number from the denominator, multiply both the numerator and the denominator by the complex conjugate of the denominator. The complex conjugate of \(-i\) is \(i\). Hence, multiply numerator and denominator by \(i\).

Step 2 - Multiply the numerator and denominator

Perform the multiplication: \(\frac{8}{-i} \times \frac{i}{i} = \frac{8i}{-i^2}\).

Step 3 - Simplify the expression

Simplify the denominator first. Recall that \(i^2 = -1\), thus: \(-i^2 = -(-1) = 1\). So the expression becomes: \frac{8i}{1} = 8i\.

Step 4 - Write in standard form

The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. For this problem, it simplifies directly to: \(0 + 8i\).

Key Concepts

Standard Form

When dealing with complex numbers, we often need to express our answers in a specific format called the standard form. The standard form of a complex number is written as \(a + bi\), where \(a\) and \(b\) are real numbers. The part \(a\) represents the real component, and \(bi\) represents the imaginary component.
For example, in the expression \(0 + 8i\), the real part is \(0\), and the imaginary part is \(8i\).
To write any complex number in standard form, simply separate its real and imaginary parts and ensure it fits the \(a + bi\) template. This form is essential because it allows us to easily compare and manipulate complex numbers.

Complex Conjugate

In the solution, we multiply by the complex conjugate to eliminate the imaginary number from the denominator. The complex conjugate of a complex number is found by changing the sign of its imaginary part. So, if you have a complex number \(a + bi\), its complex conjugate is \(a - bi\).
For example, the complex conjugate of \(-i\) is \(i\).
This trick is useful because multiplying a number by its conjugate results in a real number (since the imaginary parts cancel out).
Multiplying \([-i \times i]\) results in \(-i^2 = 1\), transforming the denominator into a real number, making it easier to simplify the expression.

Imaginary Number

An imaginary number is a number that, when squared, gives a negative result. In mathematics, the most common imaginary unit is represented by \(i\), which is defined as \(i^2 = -1\). Imaginary numbers are essential in defining complex numbers, which include a real part and an imaginary part.
In this exercise, \(i\) is used as the imaginary part, and it helps in simplifying expressions with complex numbers. When you see \(i\), remember that it essentially represents the square root of \(-1\).
This properties allows us to manipulate problems and express solutions that would otherwise be impossible with only real numbers.