Problem 59 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The simplified expression is \(\frac{1}{5} + \frac{3}{5}i\).

Step 1: Identify the Complex Conjugate

The denominator is a complex number: \(3 + i\). The complex conjugate of this number is \(3 - i\). We'll multiply both the numerator and the denominator by this conjugate to simplify.

Step 2: Multiply Numerator and Denominator

Multiply the numerator \(2i\) by the complex conjugate \(3 - i\) and likewise multiply the denominator \(3 + i\) by \(3 - i\):\[\frac{2i(3-i)}{(3+i)(3-i)}\]

Step 3: Expand the Numerator

Use the distributive property (also known as FOIL for binomials) to simplify the numerator:\[2i \times 3 - 2i \times i = 6i - 2i^2\]Since \(i^2 = -1\), substitute to get:\[6i - 2(-1) = 6i + 2\]

Step 4: Expand the Denominator

Apply the difference of squares formula to the denominator \((3+i)(3-i)\):\[ (3)^2 - (i)^2 = 9 - i^2 \]Substitute \(i^2 = -1\) to simplify further:\[9 - (-1) = 9 + 1 = 10\]

Step 5: Write the Simplified Fraction

Combine the simplified components to rewrite the fraction:\[\frac{6i + 2}{10}\]Separate the terms into real and imaginary parts:\[\frac{2}{10} + \frac{6i}{10}\]

Step 6: Simplify Each Term

Simplify each part of the expression by dividing:\[\frac{2}{10} = \frac{1}{5} \quad \text{and} \quad \frac{6i}{10} = \frac{3i}{5}\]Thus, the simplified form is:\[\frac{1}{5} + \frac{3}{5}i\]

Key Concepts

Complex Conjugate

To understand complex conjugates, start with a complex number, which often looks like this: \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The complex conjugate is formed by changing the sign of the imaginary part, resulting in \(a - bi\). This simple change plays a crucial role in simplifying complex number division.
Multiplying a complex number by its conjugate always yields a real number. This is due to a special formula, called the difference of squares:

• For any complex number \((x + yi)\), its product with its complex conjugate \((x - yi)\) is \( x^2 + y^2 \).
• This operation removes the imaginary unit \(i\) since \(i^2 = -1\), resulting in a neat and real output.

In our exercise, the complex conjugate of the denominator \(3 + i\) is \(3 - i\). By employing this technique, we transformed the division of complex numbers into an easier operation.

Division of Complex Numbers

Dividing complex numbers might seem tricky at first. However, by using the complex conjugate of the denominator, we can simplify to achieve a straightforward result.
Here's how it works:

• The numerator and denominator are each multiplied by the conjugate of the denominator.
• For the expression \(\frac{2i}{3+i}\), both parts get multiplied by \(3-i\).

This maneuver doesn't change the value of the original expression but cleverly rearranges it to where our problematic denominator has transformed into a real number. With this application, reducing the complex fraction \(\frac{2i(3-i)}{(3+i)(3-i)}\) becomes straightforward, as we'll see in the simplification process.

Simplification

Once the division is reformatted, simplification becomes an accessible task.
We proceed by simplifying separately the newly structured equation's numerator and denominator.For the numerator, \(2i(3-i)\), apply the distributive property:- \(2i \times 3 = 6i\)- \(2i \times (-i) = -2i^2\)Remember, \(i^2\) equates to \(-1\), transforming \(-2i^2\) to \(2\). So, the numerator, \(6i - 2i^2\), simplifies to \(6i + 2\).
For the denominator, \((3+i)(3-i)\), use the difference of squares:- \(3^2 - i^2 = 9 - (-1) = 10\)This simplified quotient \(\frac{6i+2}{10}\) is then separated into individual components:

• Real part: \(\frac{2}{10} = \frac{1}{5}\)
• Imaginary part: \(\frac{6i}{10} = \frac{3i}{5}\)

Finally, the fraction becomes \(\frac{1}{5} + \frac{3}{5}i\), offering a clearer and simpler perspective on complex number operations.