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Chapter 2: Problem 46

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

Short Answer

Expert verified
The results of the operations are x + y = \( \frac{7}{10} + 2i\sqrt{3} \), x - y = \( \frac{3}{10} - 4i\sqrt{3} \), x * y = 3 - \(\frac{3}{10}i\), x / y = -\(\frac{2}{19}\) - \(\frac{9}{19}i\)

Step by step solution

01

Compute x + y

Add the real parts and the imaginary parts of x and y separately. \n x + y = \( (\frac{1}{2}+\frac{1}{5}) + (-i\sqrt{3}+3i\sqrt{3}) \) = \( \frac{7}{10} + 2i\sqrt{3} \)
02

Compute x - y

Subtract the real parts and the imaginary parts of y from x separately. \n x - y = \( (\frac{1}{2}-\frac{1}{5}) + (-i\sqrt{3}-3i\sqrt{3}) \) = \( \frac{3}{10} - 4i\sqrt{3} \)
03

Compute x * y

Multiply two complex numbers using the formula (a + bi)(c + di) = ac - bd + (ad + bc)i. \n x * y = \( (\frac{1}{2}*\frac{1}{5} - (-\sqrt{3} * 3\sqrt{3})) + ((\frac{1}{2}*3\sqrt{3} + -\sqrt{3}*\frac{1}{5})i) \) = 3 - \(\frac{3}{10}i\)
04

Compute x / y

Divide two complex numbers using the formula \(\frac{a+bi}{c+di}\) = \(\frac{(ac+bd)}{(c^2+d^2)} + \frac{(bc-ad)}{(c^2+d^2)}i\). \n x / y = \( \frac{(\frac{1}{2}*\frac{1}{5} + -\sqrt{3} * 3\sqrt{3})}{(\frac{1}{5}^2 + (3\sqrt{3})^2)} + \frac{(-\sqrt{3}*\frac{1}{5} - \frac{1}{2}*3\sqrt{3})} {(\frac{1}{5}^2 + (3\sqrt{3})^2)i} \) = -\(\frac{2}{19}\) - \(\frac{9}{19}i\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Operations with Complex Numbers
Complex numbers, which have both a real and an imaginary component, add a whole new dimension to algebra. The basic form of a complex number is expressed as a + bi, where a represents the real part and bi represents the imaginary part with i being the imaginary unit satisfying i^2 = -1.

Understanding operations with complex numbers is crucial for solving a myriad of problems in mathematics, particularly within the field of complex analysis, control theory, and electrical engineering. The operations include addition, subtraction, multiplication, and division, with each having specific rules to follow to ensure the correct mathematical manipulation of their real and imaginary parts.
Adding Complex Numbers
Adding complex numbers is straightforward; you align the like terms (real with real and imaginary with imaginary) and perform simple addition. For example, consider two complex numbers x = a + bi and y = c + di. Their sum is obtained by adding the real parts (a + c) and the imaginary parts (b + d) separately.

This is similar to combining like terms in algebraic expressions and leads to a new complex number z = (a + c) + (b + d)i. This operation is commutative, meaning that x + y is always equal to y + x.
Subtracting Complex Numbers
Subtracting complex numbers involves something akin to the addition process but with a subtraction sign. Again, we align similar terms: the real parts with each other, and the imaginary parts with each other. To subtract y from x, you subtract the real part of y from the real part of x and subtract the imaginary part of y from the imaginary part of x.

The result is another complex number z = (a - c) + (b - d)i. It's essential to pay attention to signs, as subtraction is not commutative, which means x - y is not the same as y - x.
Multiplying Complex Numbers
When multiplying complex numbers, we utilize the distributive property to expand the product and then combine like terms. Taking our complex numbers x = a + bi and y = c + di, the product x * y is calculated by multiplying each part of the first complex number by each part of the second.

This results in (ac - bd) + (ad + bc)i, since i^2 equals -1. Note that when multiplying, the cross-terms generate the new imaginary part of the resulting complex number. As with addition, multiplication of complex numbers is also commutative.
Dividing Complex Numbers
Dividing complex numbers is more involved than multiplication due to the necessity of rationalizing the denominator. To divide x by y (x / y), where x = a + bi and y = c + di, you multiply both the numerator and the denominator by the conjugate of the denominator.

The conjugate of a complex number c + di is c - di. By multiplying top and bottom by this conjugate, the denominator becomes a real number (since (c + di)(c - di) = c^2 + d^2), and the complex number in the numerator is simplified to a standard form. This results in a new complex number with a real part of (ac + bd) / (c^2 + d^2) and an imaginary part of (bc - ad) / (c^2 + d^2). The division is not commutative and requires careful handling to ensure accuracy.

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Most popular questions from this chapter

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