Problem 97 - A Graphical Approach to College Algebra

Question

Divide as indicated. Write each quotient in standard form. $$\frac{-10}{i}$$

Step-by-Step Solution

Answer

The quotient is \(10i\).

Step 1: Understand the Problem

To divide \(-10\) by \(i\), where \(i\) is the imaginary unit, we want to find a way to express the quotient in standard form \(a + bi\).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by \(-i\) to eliminate the imaginary unit in the denominator.\[\frac{-10}{i} \times \frac{-i}{-i} = \frac{10i}{-i^2}\]

Step 3: Simplify the Denominator

Since \(i^2 = -1\), the denominator becomes \(-(-1) = 1\):\[\frac{10i}{1} = 10i\]

Step 4: Write in Standard Form

The result \(10i\) is already in standard form \(a + bi\) with \(a = 0\) and \(b = 10\).

Key Concepts

Imaginary Unit

In the realm of complex numbers, the imaginary unit is denoted as \(i\). It is defined by the property \(i^2 = -1\). This foundational concept allows mathematicians to extend the real number system to include solutions to problems otherwise deemed unsolvable. For instance, the square root of a negative number, such as \(\sqrt{-1}\), doesn't exist in the world of real numbers, but with the help of the imaginary unit, we express it as \(i\).

• History: The imaginary unit was first introduced to solve cubic equations in the 16th century.
• Use: It is primarily used in complex numbers, which have the form \(a + bi\).

This combination of a real part \(a\), and an imaginary part \(bi\), forms the complex number system, broadening previously established mathematical boundaries.

Standard Form

Complex numbers are commonly expressed in what is known as the standard form, \(a + bi\). Here, \(a\) and \(b\) are real numbers. The part \(a\) represents the real component, while \(bi\) is the imaginary component. Writing complex numbers in standard form offers a universally accepted way to easily carry out arithmetical operations such as addition, subtraction, and multiplication.

• Real Part: This is the \(a\) in the expression \(a + bi\).
• Imaginary Part: This is the \(bi\), where \(b\) is a real number and \(i\) is the imaginary unit.

For example, in the division problem \(\frac{-10}{i}\), we aim to express the result in standard form \(a + bi\), where in this case \(a = 0\) and \(b = 10\), hence resulting in \(10i\). This standardized approach simplifies calculations and understanding.

Conjugate

In complex numbers, the term conjugate refers to the counterpart of a given complex number. If you have a complex number, \(a + bi\), its conjugate is \(a - bi\). Essentially, to get the conjugate, you change the sign of the imaginary part.

• Purpose: Using the conjugate is vital when dividing complex numbers because it helps eliminate the imaginary unit from the denominator, giving a real number instead. This is precisely what was done in the division example \(\frac{-10}{i}\), by multiplying both numerator and denominator by \(-i\).
• Properties: The multiplication of a complex number by its conjugate results in a real number \((a^2 + b^2)\).

Thus, applying the concept of conjugates ensures that complex divisions result in expressions in standard form, allowing for clear and coherent solutions.