Problem 90 - Algebra for College Students

Question

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-4-7 i}{6 i}$$

Step-by-Step Solution

Answer

The standard form of the quotient is \(-\frac{7}{6} + \frac{2}{3}i\).

Step 1: Identify the Complex Division Expression

We need to find the quotient of the expression \( \frac{-4-7i}{6i} \). In this complex division, the numerator is \(-4 - 7i\) and the denominator is \(6i\). To simplify, we'll multiply the numerator and the denominator by the complex conjugate of the denominator.

Step 2: Multiply by the Complex Conjugate

The complex conjugate of \(6i\) is \(-6i\). Multiply both the numerator and the denominator by \(-6i\):\[\frac{-4-7i}{6i} \times \frac{-6i}{-6i} = \frac{(-4-7i)(-6i)}{(6i)(-6i)}\]

Step 3: Simplify the Denominator

Calculate the product of the denominator, \((6i)(-6i) = -36i^2\). Recall that \(i^2 = -1\), so:\[-36i^2 = -36(-1) = 36\]

Step 4: Simplify the Numerator

Expand the numerator \((-4 - 7i)(-6i)\):\[(-4)(-6i) + (-7i)(-6i) = 24i + 42i^2\]Simplify using \(i^2 = -1\):\[24i + 42(-1) = 24i - 42\]

Step 5: Combine and Express in Standard Form

Combine and simplify the entire expression:\[\frac{24i - 42}{36} = \frac{-42}{36} + \frac{24i}{36} = -\frac{7}{6} + \frac{2}{3}i\]Thus, the expression in standard form is \(-\frac{7}{6} + \frac{2}{3}i\).

Key Concepts

Complex Conjugate

In the realm of complex numbers, a complex conjugate is a powerful concept used to simplify division among complex numbers. The complex conjugate of a complex number is found by changing the sign of the imaginary part. For instance, if our complex number is represented as \(a + bi\), where \(a\) and \(b\) are real numbers, its complex conjugate would be \(a - bi\).

• The complex conjugate effectively "flips" the sign of the imaginary component.
• When a complex number is multiplied by its conjugate, the result is a real number.
• This property helps in eliminating imaginary parts during division.

Let's apply this understanding to our example: The denominator is \(6i\), so its complex conjugate is \(-6i\). By multiplying the numerator and the denominator by the conjugate, we can eliminate the imaginary unit in the denominator. That way, the division simplifies more easily into standard form.

Multiplication of Complex Numbers

Multiplication of complex numbers is straightforward once you grasp the distributive property, like multiplying binomials. When multiplying two complex numbers, \((a + bi)\) and \((c + di)\), you apply the distributive property:

• Multiply each term in the first complex number by each term in the second.
• Combine like terms to simplify.

This results in:\[(a + bi)(c + di) = ac + adi + bci + bdi^2\]Recalling that \(i^2 = -1\), helps simplify the expression further. In our exercise, multiplying \((-4 - 7i)\) by \(-6i\) involves distributing each term:

• \((-4) \times (-6i) = 24i\)
• \((-7i) \times (-6i) = 42i^2\)

Given \(i^2 = -1\), we simplify \(42i^2\) to \(-42\). Combining terms gives us \(24i - 42\). This multiplication step is critical as it prepares our expression for further simplification into the standard form.

Standard Form

Complex numbers in standard form are expressed as \(a + bi\), where \(a\) and \(b\) are real numbers. The standard form is essential because it gives a clear and concise way to display complex numbers, separating the real part from the imaginary part.

• It allows comparing complex numbers easily.
• Facilitates other operations like addition and subtraction.

From our exercise solution, after simplifying both the numerator and the denominator, we combine and divide each separately by \(36\), resulting in two parts:

• Real part: \(-\frac{42}{36} = -\frac{7}{6}\).
• Imaginary part: \(\frac{24i}{36} = \frac{2}{3}i\).

Bringing these together gives us the complex number in its standard form: \(-\frac{7}{6} + \frac{2}{3}i\). Expressing results in standard form makes it simpler for further mathematical applications and comparisons.