Can someone tell me a real life use for imaginary numbers?

Question

I am looking for a real application for imaginary numbers....not to find the complex root of an equation... something that really makes them useful!

Answer 1

Hi,

Here you go:

Applications of imaginary numbers

Despite their name, imaginary numbers are as "real" as real numbers.[2] (See the definition of complex numbers on how they can be constructed using set theory.) One way to understand this is by realizing that numbers themselves are abstractions, and the abstractions can be valid even when they are not recognized in a given context. For example, fractions such as ⅔ and ⅛ are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Similarly, negative numbers such as - 3 and - 5 are meaningless when keeping score in a US football game, but essential when keeping track of monetary debits and credits[1] (or yards gained on a play in the same football game).

Imaginary numbers follow the same pattern. For most human tasks, real numbers (or even rational numbers) offer an adequate description of data, and imaginary numbers have no meaning; however, in many areas of science and mathematics, imaginary numbers (and complex numbers in general) are essential for describing reality. Imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, and cartography.

For example, in electrical engineering, when analyzing AC circuitry, the values for the electrical voltage (and current) are expressed as imaginary or complex numbers known as phasors. These are real voltages that can cause damage/harm to either humans or equipment even if their values contain no "real part". The study of AC (alternating current) entails introduction to electricity governed by trigonometric (i.e. oscillating) functions. From calculus, one knows that differentiating or integrating either "+/- sin(t)" or "+/- cos(t)" four times (with respect to "t," of course) results in the original function "+/- sin(t)" or "+/- cos(t)." From complex algebra, one knows that multiplying the imaginary unit quantity "i" by itself four times will result in the number 1 (identity). Thus, calculus can be represented by the algebraic properties of the imaginary unit quantity (this was exploited by Charles Proteus Steinmetz).

Specifically, Euler's formula is used extensively to express signals (e.g., electromagnetic) that vary periodically over time as a combination of sine and cosine functions. Euler's formula accomplishes this more conveniently via an expression of exponential functions with imaginary exponents. Euler's formula states that, for any real number x,

e^{ix} = cos( x) + isin (x).

Some programming languages also have built-in support for imaginary numbers. For example, in the Python interpreter, one may use them by appending a lowercase or uppercase J to the number:[3]

>>> (5+2j) * (8+5j)
(30+41j)

History

Descartes was the first to use the term “imaginary” number in 1637. However, the discovery of imaginary numbers occurred much earlier by Gerolamo Cardano in the 1500s but they were not widely accepted until the work of Leonhard Euler (1707-1783) and Carl Friedrich Gauss (1777-1855). In 1978 Dr. Valery Golubenko proposed his own definition of the unit of imaginary numbers in higher mathematics.

See also

* Complex number

Answer 2

I can think of two. First of all, note that (e^(ix)+e^(-x))/2 = cos(x), and there is a similar expression for sin(x). So trigonometry comes out naturally as an extension of the study of exponential functions. This implies that certain formulas that deal with sine waves that are 90 degrees apart from each other can be described with imaginary numbers. Examples would be capacitance, impedance, and resistance in electric circuit theory. These are imaginary multiples of each other, if I recall correctly.

The other would be if you wanted to construct a pentagon, like the US Department of Defense did in the 1940s. You could use geometry to figure out what the (x,y) coordinates of the vertices of a regular pentagon are, but another way is to compute the roots of x^5-1 = 0. 1 is such a root, since 1^5 = 1. There are four others. If you solve for them, you get

+/- (sqrt(5)+1)/4 +/- i * (sqrt(10+/- 2 sqrt(5))/4)

and you can compute these quantities fairly easily. If you plot them on the complex plane, you get a regular pentagon.

Answer 3

I think what you mean is an ordinary everyday application for imaginary numbers. The problem with that is that ordinary everyday non-engineering matters rarely make use of any interesting mathematics at all. Just the stuff you find in a simple calculator. But in the engineering world, the use of complex numbers is vast. What about aerospace engineering, where conformal mappings are standard fare? What about semiconductor engineering, which makes use of quantum physics which is nearly impossible without complex numbers? What about optical design which equations are greatly facilitated by the use of complex numbers when dealing with the diffraction properties of light? Electrical engineering we already know all about, phase equations which are easier to solve with complex numbers. The list is quite long, but I'll try to think of a simple "everyday" use of imaginary numbers. I know. Computer graphic artists rendering interesting and complicated Mandelbrot sets (there's dozens of websites with this kind of artwork) have got to use complex numbers because that's how those pretty pictures are generated. How about that? Even non-engineering artists sometimes use them.

Check out the real pretty pictures in the 2nd link.

Answer 4

Use Of Imaginary Numbers

Answer 5

you're saying that internet is killing society and yes i do agree with you to a certain extent. however, without internet we really wouldnt be able to function. it lets us communicate with people halfway across the world, which isn't a bad thing at all. another thing, you say that you want to stay annonymous (for your own reasons) and then you say that you don't want to face the real world because you're "just some lonely little person" that nobody can see. unless i'm misunderstanding something, you want to be annoymous and yet you don't want to be a nobody. is it just me or is that kind of an oxymoron? I'm sure you're also not the only person in the world to have perspective. everyone has times where they feel insignificant, they just obviously deal with it differently than you do. You say that you don't see the point of reality because you're just gonna die anyway. I'm sorry but you're not living the matrix here. every second of your life is your reality. and if you think like that then of course there's no point. everyone's gonna die someday so might as well do something with your life while you can rather than sulking in front of a computer all day telling others to live theirs the way you want to.

Answer 6

They are also useful for physics and navigation. Complex numbers allow us to manipulate vectors as simple poynomials. They also allow us to work with multidimensional space, beyond the usual human capacity to visualise in 3 dimensions.
If you're a grunt worker you're not likely to use complex numbers. But if you're in charge of designing or navigating or troubleshooting, you're likely to use complex numbers.

Answer 7

How about for solving steady state AC circuits in the time domain?

Answer 8

In electronics they are used, especially in creating special spaces for experimental purposes.

Answer 9

I recall that they are essential to electronics applications.

Answer 10

physics
electronics
mathematics
engineering
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and the various sub-branches of these categories.