Problem 26 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ \frac{-2+6 i}{3 i} $$

Step-by-Step Solution

Answer

The expression \(\frac{-2+6i}{3i}\) simplifies to \(2 + \frac{2}{3}i\).

Step 1: Identify the Denominator

The denominator of the expression is purely imaginary, represented by \(3i\). To eliminate the imaginary number in the denominator, we will multiply both the numerator and the denominator by its complex conjugate.

Step 2: Multiply by the Conjugate

Multiply both the numerator \(-2 + 6i\) and the denominator \(3i\) by the conjugate of the denominator, which is \(-3i\). This process is done to remove the imaginary unit from the denominator.

Step 3: Calculate the Denominator

The denominator becomes \( (3i)(-3i) = -9i^2\). Since \(i^2 = -1\), we have \(-9(-1) = 9\). The new denominator is 9.

Step 4: Expand the Numerator

Distribute the \(-3i\) across the terms in the numerator: \((-2)(-3i) + (6i)(-3i) = 6i - 18i^2\).

Step 5: Simplify the Numerator

Since \(i^2 = -1\), substitute \(-18i^2\) with \(18\). The numerator becomes \(18 + 6i\).

Step 6: Write in Standard Form

Now, we write the simplified expression \(\frac{18 + 6i}{9}\) as separate terms: \(\frac{18}{9} + \frac{6i}{9}\), simplifying to \(2 + \frac{2}{3}i\).

Key Concepts

Complex Conjugates

Complex conjugates play a crucial role when working with complex numbers, especially in simplifying expressions like the one in the given exercise. A complex conjugate is formed by changing the sign of the imaginary part of a complex number. So, if you have a complex number of the form \(a + bi\), its complex conjugate would be \(a - bi\).
For example, let's consider the imaginary term \(3i\). Its complex conjugate would be \(-3i\).
When dealing with fractions with complex numbers, one important step is multiplying the numerator and the denominator by the conjugate of the denominator. This process helps eliminate the imaginary unit \(i\) from the denominator, making it possible to simplify the expression into a standard form \(a + bi\).

• Conjugates help in rationalizing denominators.
• Multiplying complex conjugates results in a real number since \(i^2 = -1\).

Imaginary Numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit \(i\), where \(i^2 = -1\). While they may initially seem abstract, imaginary numbers serve many practical purposes, particularly in engineering and physics.
In complex numbers, we denote such numbers with a term such as \(bi\), where \(b\) is a real number. If you see \(3i\) in an expression, it means \(3\times i\), referring to \(3\times \sqrt{-1}\).

• Imaginary numbers can be purely imaginary (like \(3i\)) or part of a complex number (like \(2 + 3i\)).
• They are essential in handling square roots of negative numbers, which cannot be solved by real numbers alone.
• In calculations, it's necessary to remember that \(i^2 = -1\), which simplifies the math involving complex numbers.

Real and Imaginary Parts

Every complex number consists of a real part and an imaginary part. The real part of a complex number \(a + bi\) is \(a\), while the imaginary part is \(bi\).
Understanding how to separate these two components is vital when simplifying complex expressions. For example, if a complex number results from simplifying an expression such as \(\frac{18 + 6i}{9}\), it can be rewritten by separating these parts as \(\frac{18}{9} + \frac{6i}{9}\) to give \(2 + \frac{2}{3}i\).

• The real part (\(a\)) is without the imaginary unit \(i\), while the imaginary part includes \(i\).
• Simplifying expressions into the form \(a + bi\) makes them easier to work with and compare.
• In calculations and equations, separating the real and imaginary parts is often a necessary step because equations involving complex numbers typically require handling each part separately.