Problem 89 - Algebra for College Students

Question

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{-2+6 i}{3 i}$$

Step-by-Step Solution

Answer

2 + \(\frac{2}{3}i\)

Step 1: Identify Numerator and Denominator

The problem presents a complex fraction \( \frac{-2+6i}{3i} \). The numerator is \(-2 + 6i\) and the denominator is \(3i\).

Step 2: Multiply the Numerator and Denominator by the Conjugate

Since the denominator is \(3i\), multiply both the numerator and the denominator by \(-i\), the conjugate of \(i\), to remove the imaginary unit from the denominator. This transforms the expression into \( \frac{(-2+6i)(-i)}{3i(-i)} \).

Step 3: Simplify the Denominator

Calculate the new denominator: \[ 3i \times (-i) = -3i^2. \]Since \(i^2 = -1\), this becomes: \[ -3(-1) = 3. \] Thus, the denominator simplifies to 3.

Step 4: Distribute in the Numerator

Distribute \(-i\) across \(-2 + 6i\) in the numerator:\[(-2)(-i) + (6i)(-i) = 2i - 6i^2.\]Simplify this by substituting \(i^2 = -1\):\[2i - 6(-1) = 2i + 6.\]So the numerator becomes \(6 + 2i\).

Step 5: Write in Standard Form

Express the fraction \(\frac{6 + 2i}{3}\) in standard form. Divide both the real and imaginary parts by 3:\[\frac{6}{3} + \frac{2i}{3} = 2 + \frac{2}{3}i.\]Thus, the solution in standard form is \(2 + \frac{2}{3}i\).

Key Concepts

Standard Form of a Complex Number

Complex numbers are a combination of a real part and an imaginary part. The standard form of a complex number is expressed as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) represents \(\sqrt{-1}\).

For example, in the complex number \(-2 + 6i\), \(-2\) is the real part, and \(6i\) is the imaginary part. The standard form helps you easily distinguish between these two parts by clearly marking the real number from the imaginary number.

Expressing a complex number in standard form is useful because it allows for straightforward calculations involving addition, subtraction, multiplication, and division with other complex numbers. When dealing with complex fractions, as in this exercise, converting them to standard form ensures clarity and simplicity in further operations.

Complex Fraction

A complex fraction, as the term suggests, is a fraction where the numerator or denominator (or both) contain complex numbers. Solving complex fractions involves a few steps that aim to simplify the expression while eliminating any imaginary units from the denominator.

To tackle a complex fraction like \(\frac{-2+6i}{3i}\), you typically multiply both the numerator and the denominator by a number that will help get rid of the imaginary unit in the denominator. This is a strategic move to make the fraction easier to handle and to convert it into a more manageable form.

The final goal is to transform the expression into a format where both parts of the final complex number are clear and simplified, typically taking the standard form \(a + bi\). This approach clarifies the real and imaginary components of the quotient.

Conjugate of Complex Numbers

The conjugate of a complex number is a tool used to eliminate the imaginary unit from the denominator in complex fractions. Given a complex number \(a + bi\), its conjugate is \(a - bi\). The key idea is that multiplying a complex number by its conjugate results in a real number.

In this particular exercise, the denominator was \(3i\). The conjugate of \(i\) is \(-i\), so both the numerator and the denominator were multiplied by \(-i\) to simplify the fraction.

• This resulted in the simplified fraction: \(\frac{6 + 2i}{3}\), with no imaginary unit remaining in the denominator.

Utilizing the conjugate helps achieve the standard form and is fundamental when working with complex numbers.

By understanding and applying the concept of conjugates, you can solve complex mathematical problems involving division with complex numbers more effectively.