Problem 43 - Precalculus : Building Concepts and Connections

Question

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=\frac{1}{2}-3 i ; y=\frac{1}{5}+\frac{4}{3} i$$

Step-by-Step Solution

Answer

The answers are: \(x+y = \frac{7}{10} - \frac{5}{3}i\), \(x-y = \frac{3}{10} - \frac{13}{3}i\), \(xy = 4.1 + \frac{3}{5}i\) and \(x/y = -\frac{237}{482}-\frac{231}{482}i\).

Step 1: Addition

The first task is to add the two numbers \(x\) and \(y\). So it would be done like: \[(\frac{1}{2}-3i) + (\frac{1}{5}+\frac{4}{3}i) = \frac{1}{2} + \frac{1}{5} + (-3 + \frac{4}{3})i = \frac{7}{10} + \frac{-5}{3}i.\] The result of the summation process is \(x+y = \frac{7}{10} - \frac{5}{3}i\)

Step 2: Subtraction

The next task is to subtract \(y\) from \(x\): \[(\frac{1}{2}-3i) - (\frac{1}{5}+\frac{4}{3}i) = \frac{1}{2} - \frac{1}{5} + (-3 - \frac{4}{3})i = \frac{3}{10} - \frac{13}{3}i.\] In that case, \(x-y = \frac{3}{10} - \frac{13}{3}i\).

Step 3: Multiplication

Now, to multiply \(x\) and \(y\), we follow the distributive property of multiplication as in the real numbers case: \[(\frac{1}{2}-3i) * (\frac{1}{5}+\frac{4}{3}i) = \frac{1}{10} + 2i - \frac{3}{5}i - 4i^2 = \frac{1}{10} +\frac{3}{5}i + 4 = 4.1 + \frac{3}{5}i.\] So, \(x*y = 4.1 + \frac{3}{5}i\).

Step 4: Division

Finally, to divide \(x\) and \(y\), we need to multiply both the numerator and denominator by the conjugate of denominator: \[\frac{x}{y} = \frac{\frac{1}{2}-3i}{\frac{1}{5}+\frac{4}{3}i} * \frac{\frac{1}{5}-\frac{4}{3}i}{\frac{1}{5}-\frac{4}{3}i} = \frac{-\frac{79}{150}-\frac{77}{150}i}{\frac{1}{25}+\frac{16}{9}} = -\frac{79}{150}-\frac{77}{150}i \div \frac{241}{225} = -\frac{237}{482}-\frac{231}{482}i.\] Consequently, \(x/y = -\frac{237}{482}-\frac{231}{482}i\).

Key Concepts

Complex Number Addition

When we perform addition with complex numbers, it's much like combining like terms in algebra. A complex number is made up of a real part and an imaginary part. For instance, with two complex numbers such as \(a + bi\) and \(c + di\), their sum is obtained by adding the real parts and the imaginary parts separately.

Let's apply this to an example with \(x = \frac{1}{2} - 3i\) and \(y = \frac{1}{5} + \frac{4}{3}i\). Here, the real parts are \(\frac{1}{2}\) and \(\frac{1}{5}\), and the imaginary parts are \(-3i\) and \(\frac{4}{3}i\). Adding the real parts gives us \(\frac{1}{2} + \frac{1}{5} = \frac{7}{10}\), and adding the imaginary parts but ignoring the 'i' at first, \(-3 + \frac{4}{3}\), results in \(-\frac{5}{3}i\). Therefore, the sum of these complex numbers is \(\frac{7}{10} - \frac{5}{3}i\).

• It's crucial to remember to handle the imaginary unit 'i' correctly; it behaves just like a variable.
• Always combine real with real, and imaginary with imaginary.

Complex Number Subtraction

When it comes to complex number subtraction, the procedure mirrors that of addition, but this time, we subtract the corresponding parts. Given our earlier complex numbers \(x = \frac{1}{2} - 3i\) and \(y = \frac{1}{5} + \frac{4}{3}i\), the process is straightforward. Subtract the real parts: \(\frac{1}{2} - \frac{1}{5} = \frac{3}{10}\), and then subtract the imaginary parts: \(-3 - \frac{4}{3} = -\frac{13}{3}i\). This leaves us with \(x - y = \frac{3}{10} - \frac{13}{3}i\).

Take note:

• Be mindful of the signs when subtracting, especially with the imaginary parts.
• The subtraction of complex numbers can result in positive or negative imaginary parts.

Complex Number Multiplication

Multiplying complex numbers follows the rules of the distributive property, just like with polynomials. For example, if we have \(x = \frac{1}{2} - 3i\) and \(y = \frac{1}{5} + \frac{4}{3}i\), the multiplication involves distributing each term in the first complex number by each term in the second complex number.

Carrying out this multiplication: \(\left(\frac{1}{2} - 3i\right) \times \left(\frac{1}{5} + \frac{4}{3}i\right)\), gives us four terms when we expand it. Don't forget that \(i^2 = -1\), which simplifies our work. So after combining and simplifying, we get the product \(x \times y = 4.1 + \frac{3}{5}i\).

• Always remember to use the fact that \(i^2 = -1\) to simplify the terms after multiplication.
• The resulting product will have a real part and an imaginary part just like any complex number.

Complex Number Division

Dividing complex numbers can be tricky, but it becomes easier when we use the concept of the conjugate. The conjugate of a complex number \(a + bi\) is \(a - bi\). To divide \(x = \frac{1}{2} - 3i\) by \(y = \frac{1}{5} + \frac{4}{3}i\), we multiply both the numerator and the denominator by the conjugate of the denominator.

The calculation is \(\frac{x}{y} = x \times \frac{1}{y}\), which involves complex multiplication followed by a simplification where the denominator becomes a real number. After applying this method, we find that \(x/y = -\frac{237}{482}-\frac{231}{482}i\).

• Multiplying by the conjugate turns the complex denominator into a real number, making the division straightforward.
• This operation will result in a complex number as well.