Problem 43 - College Algebra with Modeling and Visualization

Question

Write the expression in standard form. $$ \frac{2 i}{10-5 i} $$

Step-by-Step Solution

Answer

The expression in standard form is \(-\frac{2}{25} + \frac{4}{25} i\).

Step 1: Identify the Conjugate

The first step is to identify the conjugate of the denominator. The denominator of the expression is \(10 - 5i\). The conjugate is obtained by changing the sign of the imaginary part, which gives us \(10 + 5i\).

Step 2: Multiply Numerator and Denominator by the Conjugate

To remove the imaginary part from the denominator, we multiply both the numerator and the denominator by the conjugate \(10 + 5i\). This gives: \(\frac{2i (10 + 5i)}{(10 - 5i)(10 + 5i)}\).

Step 3: Simplify the Denominator Using Conjugate Multiplication

The product of a complex number with its conjugate is given by \((a + bi)(a - bi) = a^2 + b^2\). Apply this: \((10)^2 + (5)^2 = 100 + 25 = 125\). The denominator simplifies to 125.

Step 4: Distribute in the Numerator

Expand the numerator: \(2i(10 + 5i) = 20i + 10i^2\). Remembering that \(i^2 = -1\), substitute to get \(20i - 10\).

Step 5: Write in Standard Form

Arrange \(20i - 10\) in the form \(a + bi\), where \(a = -10\) and \(b = 20\). Divide each part by the real number in the denominator, 125. This results in \(-\frac{10}{125} + \frac{20i}{125}\). Simplify: \(-\frac{2}{25} + \frac{4i}{25}\).

Step 6: Present the Final Solution

The expression in standard form is \(-\frac{2}{25} + \frac{4}{25} i\).

Key Concepts

Standard Form

Standard form for complex numbers is crucial for clarity when working with them. A complex number is typically represented in the form \(a + bi\), where \(a\) and \(b\) are real numbers. The term \(a\) is the real part, and \(bi\) is the imaginary part.
For the complex number to be in standard form:

• \(a\) should always appear before \(bi\)
• The imaginary part \(b\) can be zero and is followed by an \(i\)

Writing expressions in standard form makes it easier to compare complex numbers or perform arithmetic operations. For example, if you convert the expression \(-\frac{2}{25} + \frac{4}{25}i\) into such a form, it is mathematically ready for further calculations.

Conjugate

In complex number arithmetic, the conjugate of a complex number is an important tool. The conjugate of a complex number \(a + bi\) is \(a - bi\).
Finding a conjugate is simple:

• The real part \(a\) remains unchanged.
• The sign of the imaginary part \(b\) is reversed.

This is particularly useful in dividing complex numbers, as multiplying a complex number by its conjugate eliminates the imaginary part in the denominator. For example, when dividing by \(10 - 5i\), multiplying by its conjugate \(10 + 5i\) simplifies the operation, resulting in a real number denominator.

Imaginary Unit

The imaginary unit is denoted by \(i\), and is defined by the property that \(i^2 = -1\). It allows us to extend the real number system to include numbers that represent the square roots of negative numbers.
Understanding \(i\) involves:

• Recognizing that \(i\) is the basis for imaginary numbers.
• Remembering that any power of \(i\) cycles every four powers: \(i^1 = i, \; i^2 = -1, \; i^3 = -i, \;\) and \(i^4 = 1\).

This cyclical nature is vital for simplifying expressions with powers of \(i\), as seen in the step where \(10i^2\) simplifies to \(-10\) due to \(i^2 = -1\).

Complex Number Multiplication

Complex number multiplication involves multiplying the numbers like binomials. It is similar to performing the distributive property over two binomial expressions \((a + bi)(c + di)\).
Here's a simple approach to remember:

• Multiply the real parts together and the imaginary parts with each other.
• Use \(i^2 = -1\) to simplify terms whenever two imaginary parts multiply.

Applying this concept, multiply the numerator and the denominator by the conjugate \(10 + 5i\), and simplify each part separately as shown in steps 3 and 4 of the original solution. This results in an expression that can later be written in standard form. This approach helps in making complex division manageable by converting it into familiar arithmetic with real numbers.