Problem 75 - Precalculus Student Solutions Manual 5th

Question

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{-5}{i}$$

Step-by-Step Solution

Answer

The quotient is \5i\.

Step 1: Eliminate the imaginary unit from the denominator

To simplify \(\frac{-5}{i}\), we need to eliminate the imaginary unit from the denominator. Multiply both the numerator and the denominator by \(i\).

Step 2: Multiply the numerator and denominator

Multiply \(-5 \cdot i \) and \(i \cdot i \). This gives \(\frac{-5i}{i^2}\).

Step 3: Simplify using the value of \(i^2\)

Recall that \(i^2 = -1\). Thus, \(\frac{-5i}{i^2} = \frac{-5i}{-1}\).

Step 4: Final simplification

Simplify \(\frac{-5i}{-1}\) to get \(5i\).

Key Concepts

imaginary unit

The imaginary unit, represented by the symbol \(i\), is a fundamental concept in complex numbers. It is defined by the property that \(i^2 = -1\). This property sets it apart from real numbers and allows for the extension of the number system to include complex numbers.

The imaginary unit helps us express numbers that cannot be rooted in the real number system, such as the square root of negative numbers. For example, the square root of \(-1\) is written as \(i\). Here are some key points to remember:

• \(i\) is used to represent the square root of \(-1\)
• Any power of \(i\) can be simplified using the fact that \(i^2 = -1\)

Understanding \(i\) is crucial for working with complex numbers, as it forms the basis of the imaginary part of these numbers.

standard form of complex numbers

Complex numbers are expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This format is referred to as the standard form of complex numbers.

In this format:

• \(a\) is the real part of the complex number
• \(bi\) is the imaginary part of the complex number

For example, in the number \(3 + 4i\), \(3\) is the real part, and \(4i\) is the imaginary part. Writing complex numbers in standard form allows for easier operations, such as addition, subtraction, and division. It also helps in visualizing these numbers in a two-dimensional plane, known as the complex plane.

simplifying division

When dividing complex numbers, the goal is to write the result in the standard form \(a + bi\). For this, the imaginary unit in the denominator needs to be eliminated. This simplifies the complex number and makes it easier to understand and use.

To simplify a division like \(\frac{-5}{i}\), follow these steps:

• Multiply both the numerator and the denominator by \(i\). This eliminates the imaginary unit from the denominator.
• Multiply the numerator and the denominator: \(\frac{-5i}{i^2}\).
• Use the property \(i^2 = -1\) to simplify: \(\frac{-5i}{-1} = 5i\).

This process ensures the result is in the standard form, making it easier to interpret and apply. Practice these steps to become comfortable with the division and simplification of complex numbers.