Problem 92 - Algebra for College Students

Question

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{3}{10 i}$$

Step-by-Step Solution

Answer

\(0 - \frac{3}{10}i\)

Step 1: Convert Denominator to Real Number

To eliminate the imaginary unit in the denominator, we multiply both the numerator and denominator by the imaginary unit, \(i\). Thus, we have \(\frac{3}{10i} \times \frac{i}{i} = \frac{3i}{10i^2}\).

Step 2: Simplify Using \(i^2 = -1\)

Remember that \(i^2 = -1\). So, the expression becomes \(\frac{3i}{10(-1)}\). This simplifies to \(\frac{-3i}{10}\).

Step 3: Express in Standard Form

Complex numbers in standard form are written as \(a + bi\). Thus, \(\frac{-3i}{10}\) can be written as \(0 - \frac{3}{10}i\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted as \(i\), is a fundamental building block of complex numbers. It is defined such that \(i^2 = -1\). This means that \(i\) is the square root of \(-1\), which cannot be solved using only real numbers.

To work with \(i\) effectively, it's important to remember:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\) (and this pattern repeats every four powers)

This cyclical pattern helps simplify expressions involving \(i\), especially when dividing complex numbers, as seen in the solved exercise.

Standard Form

Complex numbers are expressed in the standard form as \(a + bi\), where \(a\) and \(b\) are real numbers. Here, \(a\) represents the real part, and \(bi\) represents the imaginary part.

Understanding the standard form is crucial because it allows for the proper handling and visualization of complex numbers on the complex plane. To express a complex number like \(\frac{-3i}{10}\) in this form:

• Determine the real part. In the example, the real part is \(0\).
• Determine the imaginary part. The coefficient of \(i\) is \(-\frac{3}{10}\), making it the imaginary part.

Thus, the standard form of the expression is \(0 - \frac{3}{10}i\), which succinctly separates the real and imaginary components for clarity.

Quotient of Complex Numbers

A quotient of complex numbers may initially seem daunting due to the presence of the imaginary unit in the denominator. However, a standard approach simplifies this.
To remove \(i\) from the denominator and achieve standard form:

• Multiply the numerator and denominator by \(i\). This changes \(i^2\) to \(-1\), transforming the imaginary part into a real number.
• In our exercise, \(\frac{3}{10i} \times \frac{i}{i} = \frac{3i}{10i^2}\), then simplifies to \(\frac{3i}{-10}\).
• The result \(\frac{-3i}{10}\) converts into the standard form \(0 - \frac{3}{10}i\), isolating the imaginary portion in the coefficient of \(i\).

This method ensures that the complex number is easy to work with in further calculations or graphically plotting on the complex plane.