Problem 91 - Algebra for College Students

Question

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{2}{7 i}$$

Step-by-Step Solution

Answer

The quotient is \(-\frac{2}{7}i\) in standard form.

Step 1: Identify the Complex Number Format

The expression given is \(\frac{2}{7i}\). Here, \(2\) is a real number numerator and \(7i\) is an imaginary denominator. Our goal is to express this quotient in the standard form of a complex number \(a + bi\).

Step 2: Multiply the Numerator and Denominator by the Conjugate

To remove the imaginary unit from the denominator, multiply both the numerator and the denominator by \(-i\). The conjugate of \(i\) is \(-i\). So, \[\frac{2}{7i} \times \frac{-i}{-i} = \frac{-2i}{-7i^2}\].

Step 3: Simplify the Denominator Using \(i^2 = -1\)

Substitute \(i^2 = -1\) into the expression. This gives us: \[-7i^2 = -7(-1) = 7.\] So the denominator simplifies to \(7\).

Step 4: Simplify the Expression

Replace the denominator in the expression: \[\frac{-2i}{7}.\] Since the expression \(-2i\) over \(7\) is already split into real and imaginary parts, express it as \(0 - \frac{2}{7}i\).

Step 5: Write in Standard Form

The standard form of a complex number is \(a + bi\). For our expression, this is \(0 - \frac{2}{7}i\). This simplifies to \(-\frac{2}{7}i\), which is already in the standard form where \(a = 0\) and \(b = -\frac{2}{7}\).

Key Concepts

Standard Form of Complex Numbers

Complex numbers are expressed in the standard form as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The term \(bi\) includes \(b\), which is a real number, multiplied by \(i\), the imaginary unit. This form allows easy addition, subtraction, and comparison between complex numbers.

For the complex number \(0 - \frac{2}{7}i\), it is expressed in standard form by having no real component (hence \(a = 0\)) and an imaginary component \(b = -\frac{2}{7}\). Whenever handling complex numbers, ensuring they are in this form makes calculations more straightforward.

• Standard form: \(a + bi\)
• Real part \(a\), Imaginary part \(bi\)
• Example: \(0 - \frac{2}{7}i\)

Complex Number Division

Dividing complex numbers often involves removing the imaginary part from the denominator. This is achieved by multiplying both the numerator and the denominator by the conjugate of the denominator.

For the example \(\frac{2}{7i}\), multiply by \(-i\), the conjugate of \(i\), giving \(\frac{-2i}{-7i^2}\). Using the property that \(i^2 = -1\), we simplify the denominator: \(-7i^2 = 7\). Thus, the division yields \(\frac{-2i}{7}\), expressed correctly in standard form.

• Use the conjugate to rationalize denominators.
• Simplify using \(i^2 = -1\).
• Express final result in standard form: \(a + bi\).

Imaginary Unit

The imaginary unit, denoted as \(i\), is a mathematical concept where \(i^2 = -1\). It extends the real number system to include numbers that have no real equivalent. These include all multiples of the imaginary unit.

In mathematical operations involving complex numbers, using \(i\) allows for expressing roots of negative numbers, which isn't possible with just real numbers. An important property of \(i\) is that it simplifies the squares of imaginary numbers: \(i^2 = -1\), which plays a crucial role in converting expressions like \(\frac{2}{7i}\) into a standard form.

• \(i\) allows square roots of negative numbers.
• Fundamental property: \(i^2 = -1\).
• Use \(i\) to simplify and form complex numbers.