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Chapter 2: Problem 41

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

Short Answer

Expert verified
The solution for \(x+y, x-y, x*y, x/y\) are \(7 - 3i, 1 - 7i, 22 - 7i, 22/13 + 16i/13 \) respectively.

Step by step solution

01

Find x+y

We add real parts to real parts and imaginary parts to imaginary parts so, \(x + y = (4 + 3) + (-5i + 2i) = 7 - 3i\)
02

Find x-y

We subtract the real part of y from the real part of x and the imaginary part of y from the imaginary part of x so, \(x - y = (4 - 3) + (-5i - 2i) = 1 - 7i\)
03

Find x*y

Here we multiply the complex numbers using the distributive property say \(x*y = (4-5i)(3+2i) = 12 + 8i - 15i -10(i^2) = 12 - 7i +10 = 22 - 7i - i^2 = -1\) from this \(x*y = 22 - 7i\)
04

Find x/y

First, we need to create the conjugate of y as \(y' = 3-2i\), then multiply numerator and denominator by \(y'\) to simplify denominator. So \(x/y = (4-5i)/(3+2i) = (4-5i)*(3-2i) / (3+2i)*(3-2i) = (12+10+10i+6i)/ (9+4) = (22+16i)/13 = 22/13 + 16i/13\). Thus, \(x/y = 22/13 + 16i/13\)

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Key Concepts

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

Complex Number Addition
Let's start with the basics: complex number addition combines the real parts and the imaginary parts separately. To visualize this, take two complex numbers, such as \(x = 4 - 5i\) and \(y = 3 + 2i\). When we want to find the sum \(x + y\), simply add the real numbers together (4 and 3), and then the imaginary numbers (-5i and 2i). It's like combining like terms in an algebraic expression.
For our example, the calculation goes like this: \(x + y = (4 + 3) + (-5i + 2i) = 7 - 3i\). The resulting complex number is \(7 - 3i\), which represents the combined position on the complex plane. There's no complicated twisting or turning; it's a straightforward, piece-by-piece addition.
Complex Number Subtraction
Subtraction is similar to addition, but instead of combining the like parts, we're taking the difference. For complex numbers \(x = 4 - 5i\) and \(y = 3 + 2i\), we subtract the real part of \(y\) from \(x\), and likewise for the imaginary parts.
Here's how we do it: \(x - y = (4 - 3) + (-5i - 2i) = 1 - 7i\). We're essentially moving in the opposite direction on the complex plane for each part, leading to a new complex number, \(1 - 7i\), which is the difference of \(x\) and \(y\). Subtraction in the complex realm isn't much harder than in the real world, as long as you remember to treat the real and imaginary parts distinctly.
Complex Number Multiplication
Things get a bit more interesting when we multiply complex numbers. We use the distributive property, similar to expanding \((a + b)(c + d)\), but with an extra twist. The twist lies in the fact that \(i^2 = -1\), which affects our multiplication.
Multiplication Example:
Let's multiply \(x = 4 - 5i\) by \(y = 3 + 2i\). We distribute as follows: \(x*y = (4-5i)(3+2i) = 12 + 8i - 15i - 10(i^2)\). Replacing \(i^2\) with \(-1\), and simplifying, leads to \(22 - 7i\), the product of \(x\) and \(y\). The real parts and the multiple of the imaginary parts interact to create a new position on the complex plane that represents the product.
Complex Number Division
Complex number division can seem daunting, but a systematic approach makes it manageable. To divide \(x\) by \(y\), we can鈥檛 divide directly due to the complex denominator. We need a real denominator, and for that, we use the complex conjugate.
To find \(x/y = (4-5i)/(3+2i)\), we multiply both numerator and denominator by the conjugate of \(y\), which is \(3 - 2i\). This gets rid of the imaginary part in the denominator:
\[(4-5i)(3-2i) / (3+2i)(3-2i) = (22+16i)/13\].
After simplification, we get \(22/13 + 16i/13\). Division in the complex world is about making the complex simple - by creating a real denominator, we can easily find the quotient.

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Most popular questions from this chapter

Let \(f(x)=a x+b\) and \(g(x)=c x+d,\) where \(a, b, c,\) and \(d\) are constants. Show that \((f+g)(x)\) and \((f-g)(x)\) also represent linear functions.

When designing buildings, engineers must pay careful attention to how different factors affect the load a structure can bear. The following table gives the load in terms of the weight of concrete that can be borne when threaded rod anchors of various diameters are used to form joints. $$\begin{array}{cc} \text { Diameter (in.) } & \text { Load (1b) } \\ \hline 0.3750 & 2105 \\ 0.5000 & 3750 \\ 0.6250 & 5875 \\ 0.7500 & 8460 \\ 0.8750 & 11,500 \end{array}$$ (a) Examine the table and explain why the relationship between the diameter and the load is not linear. (b) The function $$f(x)=14,926 x^{2}+148 x-51$$ gives the load (in pounds of concrete) that can be borne when rod anchors of diameter \(x\) (in inches) are employed. Use this function to determine the load for an anchor with a diameter of 0.8 inch. (c) since the rods are drilled into the concrete, the manufacturer's specifications sheet gives the load in terms of the diameter of the drill bit. This diameter is always 0.125 inch larger than the diameter of the anchor. Write the function in part (b) in terms of the diameter of the drill bit. The loads for the drill bits will be the same as the loads for the corresponding anchors. (Hint: Examine the table of values and see if you can present the table in terms of the diameter of the drill bit.)

Graph each quadratic function by finding a suitable viewing window with the help of the TABLE feature of a graphing utility. Also find the vertex of the associated parabola using the graphing utility. $$y_{1}(x)=0.4 x^{2}+20$$

For function of the form \(f(x)=\) \(a x^{2}+b x+c,\) find the discriminant, \(b^{2}-4 a c,\) and use it to determine the number of \(x\)-intercepts of the graph of \(f .\) Also determine the number of real solutions of the equation \(f(x)=0.\) $$f(x)=-x^{2}+4 x-4$$

A rectangular garden plot is to be enclosed with a fence on three of its sides and an existing wall on the fourth side. There is 45 feet of fencing material available. (a) Write an equation relating the amount of available fencing material to the lengths of the three sides that are to be fenced. (b) Use the equation in part (a) to write an expression for the width of the enclosed region in terms of its length. (c) For each value of the length given in the following table of possible dimensions for the garden plot, fill in the value of the corresponding width. Use your expression from part (b) and compute the resulting area. What do you observe about the area of the enclosed region as the dimensions of the garden plot are varied? $$\begin{array}{ccc}\hline \begin{array}{c}\text { Length } \\\\\text { (feet) }\end{array} & \begin{array}{c}\text { Width } \\\\\text { (feet) }\end{array} & \begin{array}{c}\text { Total Amount of } \\\\\text { Fencing Material (feet) }\end{array} & \begin{array}{c}\text { Area } \\\\\text { (square feet) }\end{array} \\\\\hline 5 & & 45 \\\10 & & 45 \\\15 & & 45 \\\20 & & 45 \\\30 & & 45 \\\k & & 45 \\\& &\end{array}$$ (d) Write an expression for the area of the garden plot in terms of its length. (e) Find the dimensions that will yield a garden plot with an area of 145 square feet.
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