Problem 23 - Intermediate Algebra

Question

Add or subtract as indicated. $$ \left(\frac{3}{2}+\frac{1}{3} i\right)+\left(\frac{1}{6}-\frac{3}{4} i\right) $$

Step-by-Step Solution

Answer

\( \frac{5}{3} - \frac{5}{12} i \)

Step 1: Identify Real and Imaginary Parts

To solve the given complex addition, identify the real and imaginary parts from each complex number. The real parts are \( \frac{3}{2} \) and \( \frac{1}{6} \), while the imaginary parts are \( \frac{1}{3} i \) and \( -\frac{3}{4} i \).

Step 2: Add the Real Parts

Add the real parts from both complex numbers: \( \frac{3}{2} + \frac{1}{6} \). To do this, first find a common denominator, which is 6 in this case. Convert \( \frac{3}{2} \) to \( \frac{9}{6}\), and then add it to \( \frac{1}{6} \):\[\frac{9}{6} + \frac{1}{6} = \frac{10}{6} = \frac{5}{3}\]

Step 3: Add the Imaginary Parts

Add the imaginary parts \( \frac{1}{3} i \) and \( -\frac{3}{4} i \). First, find a common denominator for the fractions, which is 12. Convert \( \frac{1}{3} i \) to \( \frac{4}{12} i \) and \( -\frac{3}{4} i \) to \( -\frac{9}{12} i \):\[\frac{4}{12} i + (-\frac{9}{12} i) = -\frac{5}{12} i\]

Step 4: Combine the Results

Now combine the results from Step 2 and Step 3 to form the final complex number: \[\frac{5}{3} - \frac{5}{12} i\]

Key Concepts

Addition of Fractions

Adding fractions is an essential skill when working with complex numbers. Fractions have numerators (top numbers) and denominators (bottom numbers). To add fractions, they must have the same denominator. This is known as a common denominator.
To find a common denominator, identify the least common multiple (LCM) of the denominators involved. For example, when adding \(\frac{3}{2}\) and \(\frac{1}{6}\), the denominators are 2 and 6. The LCM of 2 and 6 is 6.

• Change \(\frac{3}{2}\) to \(\frac{9}{6}\) by multiplying both the numerator and the denominator by 3.
• Now that both fractions have the same denominator, simply add the numerators: \(9 + 1 = 10\).
• This gives the result \(\frac{10}{6}\), which simplifies to \(\frac{5}{3}\).

This process makes adding fractions straightforward and is vital when working with complex numbers.

Imaginary Numbers

Imaginary numbers are components of complex numbers and involve the square root of negative one, denoted as \(i\). Imaginary numbers are a separate dimension from real numbers and allow for the extension of traditional arithmetic.
Each imaginary number consists of a real numerical coefficient multiplied by \(i\). For example, \(\frac{1}{3}i\) and \(-\frac{3}{4}i\) are imaginary numbers.

• The value of \(i\) is such that \(i^2 = -1\).
• When performing operations, we treat the imaginary unit \(i\) separately from the real numbers.
• Imaginary numbers can be positive or negative.

These numbers are crucial in many fields of mathematics and physics, providing deeper insights beyond the real number system.

Complex Addition

Complex addition involves combining both the real and imaginary parts of complex numbers. A complex number consists of a real part and an imaginary part, expressed as \(a + bi\).
When adding two complex numbers, like \(\left(\frac{3}{2}+\frac{1}{3}i\right)\) and \(\left(\frac{1}{6}-\frac{3}{4}i\right)\), follow these steps:

• First, add the real components: \(\frac{3}{2} + \frac{1}{6}\) which simplifies to \(\frac{5}{3}\). Use steps similar to fraction addition, finding a common denominator.
• Next, add the imaginary components: \(\frac{1}{3}i + (-\frac{3}{4}i)\). Convert these fractions to a common denominator and combine to get \(-\frac{5}{12}i\).

Finally, combine the results: \(\frac{5}{3} - \frac{5}{12}i\). Each part is calculated separately, then brought together to form the resultant complex number. This keeps complex addition systematic and straightforward.