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Magazine Chapter 2: Problem 47
Find \(x+y, x-y, x y,\) and \(x / y\). $$x=-3+i ; y=i+\frac{1}{2}$$
Short Answer
Expert verified
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 by step solution
01
Assign Values
Assign the given values to \(x\) and \(y\). So we have, \(x=-3+i\) and \(y=i+\frac 1 2\).
02
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 \(x + y = -\frac 5 2 + 2i\).
03
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, \(x - y = -\frac 7 2\).
04
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 \(-\frac 5 2 - \frac 5 2 i\).
05
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, \(x / y = 0.4 + 0.4i\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \(d\):
For example, given \(x = -3 + i\) and \(y = i + \frac{1}{2}\), add their respective parts:
- Real part: \(a + c\)
- Imaginary part: \(b + d\)
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: \(1 + 1 = 2\)
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:
Consider the variables \(x = -3 + i\) and \(y = i + \frac{1}{2}\):
- Real part: \(a - c\)
- Imaginary part: \(b - d\)
Consider the variables \(x = -3 + i\) and \(y = i + \frac{1}{2}\):
- Real part: \(-3 - \frac{1}{2} = -\frac{7}{2}\)
- Imaginary part: \(1 - 1 = 0\)
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:
Using \(x = -3 + i\) and \(y = \frac{1}{2} + i\), follow these calculations:
Multiply:
Multiplying complex numbers follows these easy distributive steps.
- Multiply each part like binomials: \((a + bi)(c + di)\)
- Distribute: \(ac + adi + bci + bdi^2\)
- Remember that \(i^2 = -1\): \(ac + (ad + bc)i - bd\) \((i^2)\)
Using \(x = -3 + i\) and \(y = \frac{1}{2} + i\), follow these calculations:
Multiply:
- \(-3 \times \frac{1}{2} = -\frac{3}{2}\)
- \(-3 \times i = -3i\)
- \(i \times \frac{1}{2} = \frac{i}{2}\)
- \(i \times i = i^2 = -1\)
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:
Applying this to \(x = -3 + i\) and \(y = \frac{1}{2} + i\), follow these calculations:
Multiply:
Dividing complex numbers is simpler through these steps using the conjugate.
Multiply:
- Numerator: \((a + bi)(c - di)\)
- Denominator: \((c + di)(c - di) = c^2 + d^2\) because \((d^2i^2) = -d^2\)
Applying this to \(x = -3 + i\) and \(y = \frac{1}{2} + i\), follow these calculations:
Multiply:
- \((-3 + i)(\frac{1}{2} - i) = -\frac{3}{2} + \frac{3i}{2} + 1 - i\)
- \((\frac{1}{2} + i)(\frac{1}{2} - i) = \frac{1}{4} + 1 = \frac{5}{4}\)
Dividing complex numbers is simpler through these steps using the conjugate.
