Problem 47 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The results of the operations between \(x\) and \(y\) are \(x + y = -\frac 5 2 + 2i\), \(x - y = -\frac 7 2\), \(x.y = -\frac 5 2 - \frac 5 2 i\), and \(x / y = 0.4 + 0.4i\).

Step 1: Assign Values

Assign the given values to \(x\) and \(y\). So we have, \(x=-3+i\) and \(y=i+\frac 1 2\).

Step 2: Find \(x + y\)

To find \(x + y\), we add the real parts and the imaginary parts separately. The real part of \(x + y\) is the sum of the real parts of \(x\) and \(y\), which is \(-3 + \frac 1 2 = - \frac 5 2\). The imaginary part is the sum of the imaginary part of \(x\) and \(y\), which is \(1 + 1 = 2\). Hence \(x + y = -\frac 5 2 + 2i\).

Step 3: Find \(x - y\)

To find \(x - y\), we subtract \(y\) from \(x\). Thus the real part of \(x - y\) is \(-3 - \frac 1 2 = - \frac 7 2\) and the imaginary part is \(1 - 1 = 0\). Hence, \(x - y = -\frac 7 2\).

Step 4: Find \(x.y\)

To find the product \(x.y\), we multiply \(x\) and \(y\) as if they were binomials, remembering that \(i^2 = -1\). So \(x.y = (-3+i)(\frac 1 2 + i) = -\frac 3 2 - 3i + \frac i 2 + i^2 = -\frac 3 2 - \frac 5 2 i - 1\), which simplifies to \(-\frac 5 2 - \frac 5 2 i\).

Step 5: Find \(x / y\)

This is equivalent to the division of \(x\) by \(y\). Divide each part of \(x\) by \(y\), so \((-3+i)/(\frac 1 2 + i)\). To simplify this form, we will first multiply and divide by the conjugate of the denominator, which is \( \frac 1 2 - i \). Thus we have, \(\frac {(-3+i)(\frac 1 2 - i)} {(\frac 1 2 + i)(\frac 1 2 - i)} = \frac {-\frac 3 2 + i\frac 3 2 + 1 - i} {\frac 1 4 + 1} = \frac {\frac 1 2 + i\frac 1 2} {\frac 5 4} = 2\frac 1 5 + 2i \frac 1 5 = 0.4 + 0.4i\). So, \(x / y = 0.4 + 0.4i\).

Key Concepts

Addition of Complex Numbers

Adding complex numbers involves a simple step-by-step process: you add their real parts and their imaginary parts separately. Consider two complex numbers, such as \(x = a + bi\) and \(y = c + di\). When adding \(x\) and \(y\), combine the real parts \(a\) and \(c\), and the imaginary parts \(b\) and \(d\):

• Real part: \(a + c\)
• Imaginary part: \(b + d\)

Thus, \(x + y = (a + c) + (b + d)i\).

For example, given \(x = -3 + i\) and \(y = i + \frac{1}{2}\), add their respective parts:

• Real part: \(-3 + \frac{1}{2} = -\frac{5}{2}\)
• Imaginary part: \(1 + 1 = 2\)

So, \(x + y = -\frac{5}{2} + 2i\). Adding complex numbers can be this simple!

Subtraction of Complex Numbers

Subtraction of complex numbers is just as straightforward as addition. The only difference is that you subtract rather than add the corresponding parts. So, if you have \(x = a + bi\) and \(y = c + di\), the subtraction \(x - y\) involves:

• Real part: \(a - c\)
• Imaginary part: \(b - d\)

Thus, \(x - y = (a - c) + (b - d)i\).

Consider the variables \(x = -3 + i\) and \(y = i + \frac{1}{2}\):

• Real part: \(-3 - \frac{1}{2} = -\frac{7}{2}\)
• Imaginary part: \(1 - 1 = 0\)

Therefore, \(x - y = -\frac{7}{2}\). See how clear and simple subtracting complex numbers can be.

Multiplication of Complex Numbers

Multiplying complex numbers becomes straightforward once you remember \(i^2 = -1\). To multiply two complex numbers \(x = a + bi\) and \(y = c + di\), you use the distributive property:

• Multiply each part like binomials: \((a + bi)(c + di)\)
• Distribute: \(ac + adi + bci + bdi^2\)
• Remember that \(i^2 = -1\): \(ac + (ad + bc)i - bd\) \((i^2)\)

So, \(x \cdot y = (ac - bd) + (ad + bc)i\).

Using \(x = -3 + i\) and \(y = \frac{1}{2} + i\), follow these calculations:

Multiply:

• \(-3 \times \frac{1}{2} = -\frac{3}{2}\)
• \(-3 \times i = -3i\)
• \(i \times \frac{1}{2} = \frac{i}{2}\)
• \(i \times i = i^2 = -1\)

Combine these: \(-\frac{3}{2} - \frac{5}{2}i - 1 = -\frac{5}{2} - \frac{5}{2}i\).
Multiplying complex numbers follows these easy distributive steps.

Division of Complex Numbers

Dividing complex numbers might seem challenging, but it becomes easy by using the conjugate. Say you have complex numbers \(x = a + bi\) and \(y = c + di\). To divide \(x\) by \(y\), multiply both the numerator and the denominator by the conjugate of the denominator \(c - di\). This is how you do it:

Multiply:

• Numerator: \((a + bi)(c - di)\)
• Denominator: \((c + di)(c - di) = c^2 + d^2\) because \((d^2i^2) = -d^2\)

Then simplify the expression, resulting in: \[ \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2} \].

Applying this to \(x = -3 + i\) and \(y = \frac{1}{2} + i\), follow these calculations:

Multiply:

• \((-3 + i)(\frac{1}{2} - i) = -\frac{3}{2} + \frac{3i}{2} + 1 - i\)
• \((\frac{1}{2} + i)(\frac{1}{2} - i) = \frac{1}{4} + 1 = \frac{5}{4}\)

Thus, the result is \(\frac{\frac{1}{2} + \frac{1}{2}i}{\frac{5}{4}} = 0.4 + 0.4i\).
Dividing complex numbers is simpler through these steps using the conjugate.