Imaginary numbers in the real world

Overview: This article examines how complex numbers of the form $$a + bi$$
are used to describe the motion of an oscillating spring with damping.

When a mass is attached to the end of a spring and then the spring is stretched down and released, we expect the mass
and spring to bob up and down. The bobbing eventually dies down and the spring-mass system comes to a rest
(see figure below Figure 1).

Figure 1

Figure 2

Imaginary Numbers in the Real World

Figure 1
Imaginary Numbers in the Real World

Figure 2
Imaginary Numbers in the Real World

If we extract just the path indicated above, and plot it on coordinate axes we have the graph of a function
(see Figure 2 below).

  • This type of function is called a damped oscillator.
  • Oscillate means to move back and forth or up and down repeatedly.
  • Damp means that the oscillations will decrease due to some kind of friction,
    ie the spring will bounce up and down less and less until it eventually stops–this “slowing down” is damping.

Imaginary number

Complex number defined by real number multiplied by imaginary unit “i”

“Imaginary Number” and “Imaginary numbers” redirect here. For the 2013 EP by The Maine, see Imaginary Numbers (EP).

… (repeats the patternfrom blue area)
i−3 = i
i−2 = −1
i−1 = −i
i0 = 1
i1 = i
i2 = −1
i3 = −i
i4 = 1
i5 = i
i6 = −1
in = im where m ≡ n mod 4

An imaginary number is a complex number that can be written as a real number multiplied by the imaginary unit i,[note 1] which is defined by its property i2 = −1.[1] The square of an imaginary number bi is −b2. For example, 5i is an imaginary number, and its square is −25. Zero is considered to be both real and imaginary.[2]

Originally coined in the 17th century by René Descartes[3] as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler (in the 18th century) and Augustin-Louis Cauchy and Carl Friedrich Gauss (in the early 19th century).

An imaginary number bi can be added to a real number a to form a complex number of the form a + bi, where the real numbers a and b are called, respectively, the real part and the imaginary part of the complex number.[4][note 2]

History

Main article: History of complex numbers
An illustration of the complex plane. The imaginary numbers are on the vertical coordinate axis.

Although Greek mathematician and engineer Hero of Alexandria is noted as the first to have conceived these numbers,[5][6] Rafael Bombelli first set down the rules for multiplication of complex numbers in 1572. The concept had appeared in print earlier, for instance in work by Gerolamo Cardano. At the time imaginary numbers, as well as negative numbers, were poorly understood and regarded by some as fictitious or useless, much as zero once was. Many other mathematicians were slow to adopt the use of imaginary numbers, including René Descartes, who wrote about them in his La Géométrie, where the term imaginary was used and meant to be derogatory.[7][8] The use of imaginary numbers was not widely accepted until the work of Leonhard Euler (1707–1783) and Carl Friedrich Gauss (1777–1855). The geometric significance of complex numbers as points in a plane was first described by Caspar Wessel (1745–1818).[9]

In 1843 William Rowan Hamilton extended the idea of an axis of imaginary numbers in the plane to a four-dimensional space of quaternion imaginaries, in which three of the dimensions are analogous to the imaginary numbers in the complex field.

With the development of quotient rings of polynomial rings, the concept behind an imaginary number became more substantial, but then one also finds other imaginary numbers such as the j of tessarines which has a square of +1. This idea first surfaced with the articles by James Cockle beginning in 1848.[10]

Geometric interpretation

90-degree rotations in the complex plane

Geometrically, imaginary numbers are found on the vertical axis of the complex number plane, allowing them to be presented perpendicular to the real axis. One way of viewing imaginary numbers is to consider a standard number line, positively increasing in magnitude to the right, and negatively increasing in magnitude to the left. At 0 on this x-axis, a y-axis can be drawn with “positive” direction going up; “positive” imaginary numbers then increase in magnitude upwards, and “negative” imaginary numbers increase in magnitude downwards. This vertical axis is often called the “imaginary axis” and is denoted iℝ,

I

{displaystyle scriptstyle mathbb {I} }

, or ℑ.

In this representation, multiplication by –1 corresponds to a rotation of 180 degrees about the origin. Multiplication by i corresponds to a 90-degree rotation in the “positive” direction (i.e.

, counterclockwise), and the equation i2 = −1 is interpreted as saying that if we apply two 90-degree rotations about the origin, the net result is a single 180-degree rotation. Note that a 90-degree rotation in the “negative” direction (i.e. clockwise) also satisfies this interpretation.

This reflects the fact that −i also solves the equation x2 = −1. In general, multiplying by a complex number is the same as rotating around the origin by the complex number's argument, followed by a scaling by its magnitude.

Square roots of negative numbers

Care must be used when working with imaginary numbers expressed as the principal values of the square roots of negative numbers. For example:[11]

6
=

36

=

(

4
)
(

9
)


4


9

=
(
2
i
)
(
3
i
)
=
6

i

2

=

6.

{displaystyle 6={sqrt {36}}={sqrt {(-4)(-9)}}
eq {sqrt {-4}}{sqrt {-9}}=(2i)(3i)=6i^{2}=-6.}

{displaystyle 6={sqrt {36}}={sqrt {(-4)(-9)}}
eq {sqrt {-4}}{sqrt {-9}}=(2i)(3i)=6i^{2}=-6.}

Sometimes this is written as:


1
=

i

2

=


1


1

=

 (fallacy) 

(

1
)
(

1
)

=

1

=
1.

{displaystyle -1=i^{2}={sqrt {-1}}{sqrt {-1}}{stackrel { ext{ (fallacy) }}{=}}{sqrt {(-1)(-1)}}={sqrt {1}}=1.}

The fallacy occurs as the equality

x
y

=

x

y

{displaystyle {sqrt {xy}}={sqrt {x}}{sqrt {y}}}

does not hold when the variables are not suitably constrained. In this case the equality does not hold as the numbers are both negative. This can be demonstrated by,


x


y

=
i

x

 
i

y

=

i

2

x

y

=

x
y

x
y

,

{displaystyle {sqrt {-x}}{sqrt {-y}}=i{sqrt {x}} i{sqrt {y}}=i^{2}{sqrt {x}}{sqrt {y}}=-{sqrt {xy}}
eq {sqrt {xy}},}

{displaystyle {sqrt {-x}}{sqrt {-y}}=i{sqrt {x}} i{sqrt {y}}=i^{2}{sqrt {x}}{sqrt {y}}=-{sqrt {xy}}
eq {sqrt {xy}},}

where both x and y are non-negative real numbers.

See also

Look up imaginary number in Wiktionary, the free dictionary.
  • Imaginary unit
  • de Moivre's formula
  • Quaternion
  • Octonion

Notes

  1. ^ j is usually used in engineering contexts where i has other meanings (such as electrical current)
  2. ^ Both the real part and the imaginary part are defined as real numbers.
See also:  Weird coordinating conjunctions: yet, for, and so

References

  1. ^
    Uno Ingard, K. (1988). “Chapter 2”. Fundamentals of Waves and Oscillations. Cambridge University Press. p. 38. ISBN 0-521-33957-X.
  2. ^ Sinha, K.C. (2008). A Text Book of Mathematics Class XI (Second ed.). Rastogi Publications. p. 11.2. ISBN 978-81-7133-912-9.

  3. ^ Giaquinta, Mariano; Modica, Giuseppe (2004). Mathematical Analysis: Approximation and Discrete Processes (illustrated ed.). Springer Science & Business Media. p. 121. ISBN 978-0-8176-4337-9. Extract of page 121
  4. ^ Aufmann, Richard; Barker, Vernon C.; Nation, Richard (2009). College Algebra: Enhanced Edition (6th ed.).

    Cengage Learning. p. 66. ISBN 1-4390-4379-5.

  5. ^ Hargittai, István (1992). Fivefold symmetry (2nd ed.). World Scientific. p. 153. ISBN 981-02-0600-3.
  6. ^ Roy, Stephen Campbell (2007). Complex numbers: lattice simulation and zeta function applications. Horwood. p. 1. ISBN 1-904275-25-7.

  7. ^ Descartes, René, Discourse de la Méthode … (Leiden, (Netherlands): Jan Maire, 1637), appended book: La Géométrie, book three, p. 380.

    From page 380: “Au reste tant les vrayes racines que les fausses ne sont pas tousjours reelles; mais quelquefois seulement imaginaires; c'est a dire qu'on peut bien tousjours en imaginer autant que jay dit en chasque Equation; mais qu'il n'y a quelquefois aucune quantité, qui corresponde a celles qu'on imagine, comme encore qu'on en puisse imaginer trois en celle cy, x3 – 6xx + 13x – 10 = 0, il n'y en a toutefois qu'une reelle, qui est 2, & pour les deux autres, quoy qu'on les augmente, ou diminue, ou multiplie en la façon que je viens d'expliquer, on ne sçauroit les rendre autres qu'imaginaires.” (Moreover, the true roots as well as the false [roots] are not always real; but sometimes only imaginary [quantities]; that is to say, one can always imagine as many of them in each equation as I said; but there is sometimes no quantity that corresponds to what one imagines, just as although one can imagine three of them in this [equation], x3 – 6xx + 13x – 10 = 0, only one of them however is real, which is 2, and regarding the other two, although one increase, or decrease, or multiply them in the manner that I just explained, one would not be able to make them other than imaginary [quantities].)

  8. ^ Martinez, Albert A. (2006), Negative Math: How Mathematical Rules Can Be Positively Bent, Princeton: Princeton University Press, ISBN 0-691-12309-8, discusses ambiguities of meaning in imaginary expressions in historical context.
  9. ^ Rozenfeld, Boris Abramovich (1988). “Chapter 10”. A history of non-euclidean geometry: evolution of the concept of a geometric space. Springer. p. 382. ISBN 0-387-96458-4.
  10. ^ Cockle, James (1848) “On Certain Functions Resembling Quaternions and on a New Imaginary in Algebra”, London-Dublin-Edinburgh Philosophical Magazine, series 3, 33:435–9 and Cockle (1849) “On a New Imaginary in Algebra”, Philosophical Magazine 34:37–47
  11. ^ Nahin, Paul J. (2010). An Imaginary Tale: The Story of “i” [the square root of minus one]. Princeton University Press. p. 12. ISBN 978-1-4008-3029-9. Extract of page 12

Bibliography

  • Nahin, Paul (1998). An Imaginary Tale: the Story of the Square Root of −1. Princeton: Princeton University Press. ISBN 0-691-02795-1., explains many applications of imaginary expressions.

External links

  • How can one show that imaginary numbers really do exist? – an article that discusses the existence of imaginary numbers.
  • In our time: Imaginary numbers Discussion of imaginary numbers on BBC Radio 4.
  • 5Numbers programme 4 BBC Radio 4 programme
  • Why Use Imaginary Numbers? Basic Explanation and Uses of Imaginary Numbers

Imaginary Numbers in the Real World

Why did the chicken cross the road? I have no idea. But I do know how the chicken crossed the road—it used math to move and perhaps rotate itself from one point to another on its journey. At least that’s how the video game version of the chicken must have crossed the road. And in this video game version, I also know why it crossed the road—because you told it to!

Today, we’re talking about real world uses of complex numbers. So why in the world am I talking about chickens and video games? Because, believe it or not, you can use complex numbers to describe the motion of a chicken or anything else in both the video game and real worlds. And you can do a bunch of other useful stuff with complex numbers, too.

Of Chickens and Position Vectors

Let’s kick things off by talking about chickens and position vectors … because that’s not weird at all! You might be wondering why I’m compelled to contemplate this combination? Well, I'm thinking about how I might go about designing a chicken crossing the road video game. And one of the important parts of building such a game is figuring out how to keep track of the positions of chickens as they confront roads and other obstacles.

One way to do this is to set up an x-y coordinate system so that the position of each chicken can be labeled with x and y values. For example, at some point in time a chicken might be standing at position x=3, y=4 in this coordinate system before moving to position x=2, y=3.

But instead of thinking about ordered pairs of coordinates to keep track of a chicken’s location, you could also imagine drawing an arrow from the origin of the coordinate system to the chicken.

This arrow is called the chicken’s position vector, and the changes in this vector over time tell you how the chicken is moving.

Of course, the x and y components of this vector are exactly the ordered pairs of points we talked about before, but this way of thinking about locations as vectors has some advantages—so it's good to keep in mind as we go along. So, how can we make our chickens and their position vectors move around?

Adding and Subtracting Complex Numbers

To understand, we first need to talk about adding and multiplying complex numbers. I know, these two things seem to have nothing in common, but stick with me for a minute and you’ll see that they are actually related.

By adding or subtracting complex numbers…we can move the chicken anywhere in the plane.

Let’s start by thinking about the complex plane. As we’ve discussed, every complex number is made by adding a real number to an imaginary number: a + b•i, where a is the real part and b is the imaginary part.

We can plot a complex number on the complex plane—the position along the x-axis of this plane represents the real part of the complex number and the position along the y-axis represents its imaginary part.

Now imagine a chicken standing at the origin—that’s the point (0, 0)—on the complex plane. If we add or subtract the real number 1, we end up at either the point 1 or –1 on the real axis.

A vector from our chicken’s starting point (the origin) to either of these points represents its new position. If we instead start back at the origin and add or subtract the complex number i, we end up at either the point i or –i on the imaginary axis.

So by adding or subtracting a real number from the complex number representing the position of our chicken, we can make it move to the right or left. By adding or subtracting a purely imaginary number from the chicken’s complex position vector, we can make it move up or down.

And by adding or subtracting complex numbers made up of both a real and an imaginary part, we can move the chicken anywhere in the plane.

Pages

Answers and Explanations — Imaginary Numbers:
Relevance to the Real World

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Answers and Explanations

Okay, now we've seen that imaginary numbers exist. However, they exist in
the context of a different number system, something different from the number
systems we are used to. The “complex numbers” that make up this system
are pairs of numbers;
do they really deserve to be called “numbers” in their
own right?

Well, remember that fractions are pairs of numbers also. They clearly
deserve to be called numbers in their own right, since they can
measure “how much” in some contexts (for instance, “I ate three
quarters of a pie”). So, the principle of considering a pair of
numbers (in this example, 3 and 4) as a number in its own right is
well established.

The fact remains, though, that complex numbers have much less
direct relevance to real-world quantities than other numbers
do. An imaginary number could not be used as a measurement of how much
water is in a bottle, or how far a kite has travelled, or how many
fingers one has.

Nonetheless, there are a few real world quantities for which complex
numbers are the natural model. The strength of an electromagnetic
field is one example.

The field has both an electric and a magnetic
component, so it takes a pair of real numbers (one for the intensity
of the electric field, one for the intensity of the magnetic field) to
describe the field strength.

This pair of real numbers can be thought
of as a complex number, and it turns out that the strange rule of
multiplication of complex numbers has relevance to the physics of an
electromagnetic field.

Although such direct applications of complex numbers to the real world
are few, their indirect applications are many. Many properties
related to real numbers only become clear when the real numbers are
thought of as sitting inside the complex number system. Therefore,
complex numbers aid in the understanding even of things that are
described by ordinary, familiar real numbers.

Jobs that Use Imaginary & Complex Numbers

Electrical engineers develop and create different types of electrical equipment for a variety of industries, from automotive to communications.

These engineers must be highly skilled in mathematics, which includes having an advanced understanding of how to use imaginary and complex numbers, as they are required in the course of making plans for new equipment.

They also often perform mathematical calculations to make sure their plans are accurate and that the resulting products will function correctly. To become an electrical engineer, you need a bachelor's degree.

Mathematician

Mathematicians may be the most obvious choice for a career that involves imaginary and complex numbers. They perform work in a number of different industries, from scientific fields to the federal government to research.

Mathematicians may also incorporate different mathematical theories or work to create new ones, and regardless of their specific roles, they will often work with imaginary and complex numbers.

To become a mathematician, you will usually need at least a master's degree in mathematics.

Physicist

Physicists are experts in the study of matter, energy, time, and space. They study the fundamental properties that make up everything in the universe, from the smallest atom to the largest galaxy.

Much of their work involves performing advanced mathematical calculations, which could involve working with imaginary and complex numbers. To become a physicist, you will usually need a Ph.D. in the field.

Statistician

Statisticians work to solve problems in the real world by collecting data through surveys, polls, and questionnaires. They then use various statistical analysis methods to analyze the data and arrive at conclusions.

They must have a high level understanding of math and statistics, including the ability to work with imaginary and complex numbers in their data analysis.

To become a statistician, you will usually need at least a master's degree in a mathematically related field.

Software Developer

Imaginary Numbers and the Real World

This is a fairy tale, but it is not far short of the truth. The way mathematicians played with words and meanings is true enough, only the names and circumstance have been changed to keep the story moving.

Many years ago, early communities took to farming and to trading in markets.

They began to record the numbers of things they owned and traded on knotted string, others cut notches in wooden sticks, yet others scratched their tallies into clay tablets.

Presently, somebody got the bright idea of creating symbols for each number so you didn't have to count and remember long tallies. It all seemed very natural.

But something was missing. Presently money appeared and moneylenders set up in the market too. One day a lender ran short of cash and, rather than send his client to a competitor, he borrowed some more himself from another moneylender and lent that to his client.

He looked at his tablets and wondered how to record it. He scratched a little − next to the number he owed and called it a “negative number” because he lacked it. Soon, another client paid him back just what he had borrowed but not the interest on the loan.

The account balanced neatly but still had to be kept open. If he left its entry blank it would look as if he had not done his sums yet, so he scratched a little circle in there to show that there was no net cash in his purse yet. The number zero was born.

His colleagues looked over his shoulder and were disgusted.

“You shouldn't be writing those silly minus and zero numbers down,” they complained, “it's not natural.”

“What do you mean?”

“Well, numbers naturally start at one and work up. You can't go the other way, you can't have a minus sheep, and if you haven't got any sheep then you haven't got any, it's absurd to say that you have a flock of “zero” sheep in your field. I've got a “zero” empire too but that doesn't make me an emperor.”

After much discussion, and not a few black eyes and broken drinking-vessels, everybody agreed to say that there were two kinds of numbers, natural and unnatural. But the unnatural ones proved so useful that everybody started using them.

And something else was still missing. many years later a baker was having trouble. She had thought up the idea of baking one big loaf and then selling off pieces to hungry customers. It was much quicker and easier than making lots of little bread-cakes, and not so easy for the street urchins to steal either.

Then one customer tried to be sneaky. He paid for a single piece and then tried to take the whole loaf.

“Oy! Stop that! You paid for a quarter of that loaf and that's what you get!”

“No,” he wheedled, “I paid for my lump of bread. That's the lump you have got, so that's what you sold me.”

“Don't be cheeky, I sold you a quarter of it.” She cut the piece off and gave it to him, “There, that's your piece.”

  • “No, that's a quarter of a piece, I paid for a whole piece.”
  • He had probably drunk a bit too much or something, anyway in the end they had to call on the village Elders to make peace.
  • “I bought a whole piece” declared the hungry man, “you can't walk off with a fraction of a thing, you have either got something or you haven't.”
  • “It wasn't whole, said the baker crossly, it was a quarter he paid for and a quarter he got.”

The Elders saw immediately that the baker was right, but the customer was big, strong and a little drunk. They didn't want the black eyes that the moneylenders had given each other.

After a brief, muttered conversation the Chief declared that there were two kinds of number. Whole numbers were the normal order of things, it was absurd to suggest otherwise. The big man smirked and reached for the remaining loaf.

But, the Chief continued as he pushed his staff firmly in the man's way, broken fragments yielded useful fragmentary numbers too, which may be called fractions for short. If somebody sold a quarter fraction then that was what they had sold.

The big man scowled, opened his mouth to retort, saw the Chief raise his staff meaningfully and shut it again. The Elders had spoken and that was that.

Careers That Use Complex Numbers

By Stephanie Dube Dwilson

Complex numbers have both a real number element and an imaginary number element, usually in the form a + bi. In this form, a and b are real numbers, while i is an imaginary number.

An imaginary number is the designation given for the square root of negative one.

Many careers that employ higher mathematics also use complex numbers, which can help simplify mathematical equations that would otherwise be much longer.

Electrical engineers use complex numbers frequently in their careers. Alternating currents, for example, aren't one dimensional.

Instead, AC's voltage, current and resistance alternate in both amplitude and direction, along with having changing properties in terms of phase shift and frequency.

Calculating AC circuits, instead of direct current circuits, is a more complex process made simpler by utilizing complex numbers.

Quantum physicists operate in a world where nothing is known completely precisely. A quantum particle's location isn't known exactly; rather, its location is suggested in terms of probabilities. These calculations are incredibly complex and take place on a three-dimensional rather than scalar plane. Complex numbers are necessary to make these calculations possible.

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Sales analysts may utilize complex numbers in order to make predictions and better understand the sales process. For example, when faced with sales numbers for a series of years, a sales analyst might use complex numbers to calculate regression or determine the year in which sales might reach a specific, desired number.

Economists use complex numbers in order to make profit predictions. When analyzing business cycles, complex numbers can also come into play. Complex numbers may also help an economist trying to determine the internal rate of return, such as the yield on a bond, or an economic duration. Using complex numbers can help economists better visualize results.

About the Author

With features published by media such as Business Week and Fox News, Stephanie Dube Dwilson is an accomplished writer with a law degree and a master's in science and technology journalism. She has written for law firms, public relations and marketing agencies, science and technology websites, and business magazines.

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