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Chapter 2: Problem 4

First simplify each of the following numbers to the \(x+i y\) form or to the \(r e^{i \theta}\) form. Then plot the number in the complex plane. $$i^{2}+2 i+1$$

Short Answer

Expert verified
The expression simplifies to \(2i\) and is plotted at (0, 2) on the complex plane.

Step by step solution

01

Simplify the given expression

First, simplify the given expression. The expression is:\[i^{2}+2i+1\]Recall that \(i^{2} = -1\). Replacing \(i^{2}\) with \(-1\), we get:\[-1 + 2i + 1\]
02

Combine like terms

Combine the real and imaginary parts:\[-1 + 1 + 2i = 2i\]So the simplified form is:\[2i\]
03

Express in the form \(x + iy\)

Since the number is purely imaginary, we can express it as:\[0 + 2i\]This is in the form \(x + iy\), where \(x = 0\) and \(y = 2\).
04

Plot the number on the complex plane

To plot the number \(2i\) on the complex plane, place a point at the coordinates (0, 2) where the x-coordinate is 0 (real part) and the y-coordinate is 2 (imaginary part).

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Key Concepts

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

Imaginary Numbers
Imaginary numbers are a key component of complex numbers. The imaginary unit is represented by the symbol 'i', and is defined as the square root of -1. Thus, we have the fundamental relationship that \(i^{2} = -1\). Imaginary numbers can be combined with real numbers to form complex numbers, which are expressed as \(a + bi\), where 'a' is the real part and 'b' is the imaginary part. For example, in the expression \(2i\), 2 is the coefficient of the imaginary unit 'i'.
Complex Plane
The complex plane is a two-dimensional plane used to visualize complex numbers. It has two axes: the real axis (horizontal) and the imaginary axis (vertical). Each complex number corresponds to a point on this plane. To plot \(0 + 2i\), you would place a point at the coordinates (0, 2). This represents the complex number with a real part of 0 and an imaginary part of 2. The complex plane allows for a visual representation of complex numbers, making it easier to understand their properties and relationships.
Simplification of Complex Expressions
Simplifying complex expressions involves combining like terms and using the fundamental property \(i^{2} = -1\). Let's work through the example given in the problem: \(i^{2} + 2i + 1\). First, replace \(i^{2}\) with -1, so the expression becomes \(-1 + 2i + 1\). Next, combine the real parts: -1 + 1 = 0, leaving us with \(0 + 2i\). This result is in the form \(x + iy\), where \(x = 0\) and \(y = 2\). This simplification process makes it easier to identify and work with complex numbers in their standard form.

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Most popular questions from this chapter

Express the following complex numbers in the \(x+i y\) form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others- -try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. $$\left(\frac{i \sqrt{2}}{1+i}\right)^{12}$$

Express the following complex numbers in the \(x+i y\) form. Try to visualize each complex number, using sketches as in the examples if necessary. The first twelve problems you should be able to do in your head (and maybe some of the others- -try it!) Doing a problem quickly in your head saves time over using a computer. Remember that the point in doing problems like this is to gain skill in manipulating complex expressions, so a good study method is to do the problems by hand and use a computer to check your answers. $$\frac{1}{(1+i)^{3}}$$

Find each of the following in rectangular form \(x+i y\) and check your results by computer. Remember to save time by doing as much as you can in your head. $$e^{(i \pi / 4)+(\ln 2) / 2}$$

Evaluate each of the following in \(x+i y\) form, and compare with a computer solution. $$\sin \left[i \ln \left(\frac{\sqrt{3}+i}{2}\right)\right]$$

Find one or more values of each of the following complex expressions and compare with a computer solution. $$\left(\frac{1+i}{1-i}\right)^{2718}$$
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