Problem 69 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{2}{3+i} $$

Step-by-Step Solution

Answer

The result is \(\frac{3}{5} - \frac{1}{5}i\).

Step 1: Identify the Need for a Conjugate

To simplify the expression \( \frac{2}{3+i} \), we want to eliminate the imaginary part in the denominator. This can be done by multiplying both the numerator and the denominator by the conjugate of the denominator.

Step 2: Determine the Conjugate

The conjugate of \(3 + i\) is \(3 - i\). Multiplying by this conjugate will help us simplify the denominator into a real number.

Step 3: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate: \[\left(\frac{2}{3+i}\right) \cdot \left(\frac{3-i}{3-i}\right) = \frac{2(3-i)}{(3+i)(3-i)}\]

Step 4: Simplify the Numerator

Calculate \(2(3-i)\):\[2 \times 3 - 2 \times i = 6 - 2i\]

Step 5: Simplify the Denominator

Calculate \((3+i)(3-i)\) using the difference of squares:\[(3)^2 - (i)^2 = 9 - i^2 = 9 - (-1) = 10\]

Step 6: Write the Simplified Expression

Combine the simplified numerator and denominator:\[\frac{6 - 2i}{10} = \frac{6}{10} - \frac{2i}{10}\]

Step 7: Final Simplification

Reduce the fractions:\[\frac{6}{10} = \frac{3}{5} \quad \text{and} \quad \frac{2}{10} = \frac{1}{5}\]Thus, the expression becomes:\[\frac{3}{5} - \frac{1}{5}i\]

Key Concepts

Imaginary Numbers

Imagine a number built on imagination, quite literally, that's an imaginary number for you! An imaginary number is represented by the symbol \(i\), which is the square root of -1. The peculiar thing about \(i\) is that it helps us expand the concept of numbers beyond the ones we can count. What happens when we multiply \(i\) by itself? Well, it becomes \(i^2\), which equals -1. When combined with real numbers, imaginary numbers form complex numbers, such as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary part. The presence of \(i\) allows complex numbers to be versatile tools in mathematics, especially when solving some equations that have no real number solution.

Conjugate

Conjugates of complex numbers are like special partners that help simplify mathematical expressions, particularly to get rid of imaginary components in fractions or equations. Given a complex number of the form \(a + bi\), its conjugate is \(a - bi\). Notice the only change: the sign of the imaginary part flips. Why is this helpful? When a complex number is multiplied by its conjugate, it results in a real number because the imaginary parts cancel each other out. For example, if you multiply \( (3+i) \) by its conjugate \( (3-i) \), the result is a real number: \((3+i)(3-i) = 3^2 - (i)^2 = 9 + 1 = 10\). This step is crucial to simplify expressions like \(\frac{2}{3+i}\), bringing them back to a format where the denominator is a real number.

Real Numbers

Real numbers are the backbone of most math you encounter daily. They include both rational numbers, like 1, 2, 0.5, and irrational numbers, like \(\pi\) and \(\sqrt{2}\), which cannot be expressed as simple fractions. Real numbers can be found on the number line and can be added, subtracted, multiplied, and divided (except by zero). In the realm of complex numbers, the real component is, well, the `real' part. For example, in a complex number such as \(a + bi\), \(a\) represents the real part. When we rationalize expressions with complex numbers, our goal is often to convert an imaginary component in the denominator to a real number, making the expression easier to handle and understand.

Rationalization

Rationalization is a neat mathematical trick used to clear away complex or unwanted parts of a fraction’s denominator. When dealing with complex numbers, the aim is to eliminate the imaginary element from the denominator. This is where the conjugate comes into play. By multiplying both the numerator and the denominator by the conjugate, the denominator magically turns into a real number due to the difference of squares formula: \(a^2 - b^2 = (a+b)(a-b)\). In our example, we rationalize \(\frac{2}{3+i}\) by multiplying by the conjugate \(\frac{3-i}{3-i}\). This action results in a simplified, more comprehensible form where the denominator becomes a real number \((10)\), as opposed to dealing with an unsettling imaginary denominator. Once rationalized, the entire expression, such as \(\frac{6 - 2i}{10}\), can be split into real and imaginary parts, resulting ultimately in \(\frac{3}{5} - \frac{1}{5}i\).