Problem 98 - A Graphical Approach to College Algebra

Question

Divide as indicated. Write each quotient in standard form. $$\frac{-19-9 i}{i}$$

Step-by-Step Solution

Answer

The quotient is \(-9 + 19i\).

Step 1: Express the Division Problem

The given problem is to divide \(-19 - 9i\) by \(i\). In division involving complex numbers, we multiply the numerator and the denominator by the complex conjugate of the denominator. Here, the denominator is \(i\), and its conjugate is \(-i\).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by \(-i\):\[\frac{-19 - 9i}{i} \cdot \frac{-i}{-i} = \frac{(-19 - 9i)(-i)}{i(-i)}\]

Step 3: Simplify the Denominator

Simplify the denominator:\[i \times (-i) = -i^2 = 1\] (since \(i^2 = -1\)).

Step 4: Simplify the Numerator

Distribute \(-i\) through \(-19 - 9i\):\[(-19)(-i) - (9i)(-i) = 19i + 9i^2\]Recall that \(i^2 = -1\), so this becomes:\[19i + 9(-1) = 19i - 9\].

Step 5: Write the Quotient in Standard Form

The expression from Step 4 is:\[\frac{19i - 9}{1} = -9 + 19i\].The standard form for complex numbers is \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Key Concepts

standard form

When working with complex numbers, the standard form refers to expressing a complex number as \(a + bi\). Here, **\(a\)** is the real part and **\(b\)** is the imaginary part. The imaginary unit \(i\) implies \(\sqrt{-1}\). For instance, in our exercise, the quotient is represented in standard form as \(-9 + 19i\).

• **Real part**: \(-9\)
• **Imaginary part**: \(19\)

Using the standard form makes it easier to interpret and manipulate complex numbers in equations. By simplifying to this format, complex arithmetic becomes more straightforward, especially when adding, subtracting, multiplying, or dividing complex numbers.

division

Division with complex numbers involves a procedure similar to real numbers but requires additional steps due to the presence of the imaginary unit. To divide two complex numbers, one must first eliminate the imaginary unit from the denominator by multiplying both the numerator and the denominator by the complex conjugate of the denominator.
For example, in the given exercise, dividing \(-19 - 9i\) by \(i\) starts by choosing \(-i\) as the conjugate. Multiply both the numerator and the denominator by \(-i\). This transforms the division into a simpler form, allowing the computation of the real and imaginary components separately. This modification reduces the complexity and eases the simplification.

• **Step 1**: Identify the denominator's conjugate.
• **Step 2**: Multiply through by the conjugate.

By following these steps, one effectively manages division of complex numbers, achieving simplification and an accurate result.

complex conjugate

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). It mirrors the imaginary part of the number around the real axis. Mathematically, multiplying a complex number by its conjugate results in a real number. This property is particularly useful for division, as it removes the imaginary unit from the denominator, transforming it into a real number.
In our exercise, the denominator is \(i\), whose conjugate is \(-i\). When multiplying \(i\) by \(-i\), we achieve a real number:

• \(i \times -i = -i^2 = 1\), since \(i^2 = -1\).

Removing the imaginary component simplifies calculations and aids in writing complex numbers in standard form. Thus, using the complex conjugate is a crucial step in dividing complex numbers effectively.

imaginary unit

The imaginary unit \(i\) is the cornerstone of complex numbers, representing \(\sqrt{-1}\). It extends the real number system so that equations like \(x^2 + 1 = 0\) become solvable, where \(x\) would be \(i\) or \(-i\).
**Properties of \(i\)**:

• **\(i^2 = -1\)**: The square of \(i\) is \(-1\), essential for understanding multiplication and division involving complex numbers.
• **Powers of \(i\)**: It cycles every four powers - \(i^3 = -i\), \(i^4 = 1\), and \(i^5 = i\), repeating onwards.

In the exercise, we see \(i\) as the denominator and its manipulation to express the division result in standard form. Understanding the imaginary unit aids both in simplifying complex expressions and solving algebraic equations involving complex numbers.