When would you ever have to use imaginary numbers in life?

Question

I'm learning about imaginary numbers in Algebra II right now, and I'm thinking, what's the point? If the numbers are imaginary and don't really exist, then why must we learn about them? When would we ever have to use these kinds of numbers in life, like in a job or career of some sort, if they're imaginary and don't exist?

Also, if you know how and why these numbers are used in those careers, then please do tell. Thanks.

Answer 1

Don't let the name fool you. Imaginary numbers do have real life uses. Together, the use of imaginary numbers with the real numbers to make numbers of the form a+bi form the complex numbers, to which the real numbers a+0i is simply a subset of.

In fact, the natural numbers, rational numbers, integers, and real numbers are as much of an invention as the complex numbers are.

We first started using natural numbers to quantify discrete quantities such as how many apples are in the basket. This number system serves its purpose here well, but it has its limitations.

For instance, what if you want to discriminate between having 4 apples and owing 4 apples? Or gaining 4 more apples versus losing 4 more apples? This is when we invented the integers, which can be used to quantify gains and losses of discrete values.

Even then, the integers were not sufficient in quantifying other aspects of life such as ratios. This is where fractions (rational numbers) came in.

To make matters worse, it was discovered that there are yet again holes in the rational numbers. For instance, where is the square root of 2? Nowhere. It doesn't exist in the rational numbers. So we had to "complete the field" as it were and create the real numbers, which allows nth roots of any number and the Intermediate Value Theorem (I will let the Wikipedia article go into this for you):

http://en.wikipedia.org/wiki/Intermediate_Value_Theorem

It just so happened we also ran into situations where real numbers weren't enough, so we had to invent the complex numbers.

The complex numbers are merely what mathematicians call a field extension containing an element whose square is -1. What I mean by field extension is that you can do the same things and apply the same algebraic properties with complex numbers as with real numbers, except for any ordering relations (less than, for example). The result is a more versatile number system.

Also, there is the saying that the shortest path to a real result is through the complex numbers. This means that some theorems about the real numbers takes a proof that requires use of the complex numbers.

Also, by studying complex numbers, one can discover unexpected results, such as e^(x*i) = cos x + i sin x, that have some real life application in some sciences.

Now, as if complex numbers weren't enough, there are also the Hypercomplex numbers, Quaternions, Octonions, and so on, but let's not go there. :P

So basically, imaginary numbers are imaginary, as are the complex numbers, but the real numbers, rational numbers, integers, and natural numbers are just as imaginary. We just don't think of them as imaginary because we can conceptualize physical manifestations of the latter four number systems.

For more information, try this link:

http://en.wikipedia.org/wiki/Complex_numbers#Applications

Answer 2

Other than the electronics already mentioned, they can be used in phase equations in circuits. Also, you can represent vectors with them and imagine each vector having a real and a non-real component. Then, you can add, subtract, multiply, and divide the vectors and have it mean something. As for the non-engineering applications, there are coding theories, complex analysis, modern algebra, number theory, and other less applied areas of mathematics that use them extensively. They can be the answers to some trig problems that would be very hard to do without them. DeMoivre's theorem comes to mind as a way to finding the non-real roots of 1. These can then be used to convert some circular/trig representations into complex number representations.

Answer 3

There are two things to know that are related: Polar and Rectangular forms.

The rectangular for uses a point: (1,1)
The polar form uses a radius and an angle: (sqrt 2, <45)

Engineers use these to plot waveforms, and physicists do the same. These numbers are called imaginary because you doing something that is not common: taking the square root of a negative number. However, these numbers exists because of angles. If you are not going into a scientific field, then I do not know what else they are good for, but it is good to know that they exist.

Answer 4

I'm an agronomist and I used imaginary numbers when I was studying electricity in my second and third year at university. Now that I'm working with modelling of agricultural processes, I have to deal with imaginary numbers to solve mathematical equations.

Answer 5

Suppose I save $ 1000 in first year, $ 900 in second year $ 810 in third year and so on. In other words, every year I save 90% of what I saved previous year. Now, I want to find out after how many years will I be able to save a total of $ 11000.

Let x = number of years at the end of which I would have saved $ 11,000.

As 1000, 900, 810, ... is G.P. the sum in x years is given by
1000 [ (1 - (9/10)^(x -1) ] / ( 1 - 9/10 ) and we want this to be 11000.
So 11000 = 1000 [ 1 - (0.9)^(x - 1) ] ( 1 - 0.9)
=> 11 * (0.1) = 1 - (0.9)^(x -1)
=> (0.9)^(x - 1) = - 0.10
=> (x - 1) log(0.9) = log ( - 0.10)
But as log ( -0.10) is imaginary, x is imaginary.
So, we can conclude that it is not possible to save $ 11,000 by the above scheme of saving in any number of years.

There can be many situations in life like above in which imaginary numbers can be a guide to us. I wish I were able to create a better example.

Answer 6

You are being misled by the word "Imaginary" as if such numbers are but a figment of imagination and do not obey mathematical rules.
They do obey rules and are widely used in Electrical Engineering and other Engineering branches such as Mechanical and Civil not to mention Physics.
My advice would be to get to grips with complex (imaginary) numbers---they are really quite logical and conform to Mathematical rules.
They DO exist----go to it!

Answer 7

Imaginary numbers as much imaginary as irrational numbers or negative numbers. In 14th century they can not imagine negative numbers so they were "imaginary" for them but we use it regularly.
Imaginary numbers are used for example in Digital Signal Processing to describe waves and oscillations (all other technologies like vibration, noise cancellation and so on also use them extensively, as well as all heat transfer and other wave propagation applications) and without them you wouldn't have computers, radios, TVs, game consoles, cellphones, etc.
If you are going to be administrative assistant you would not need no rational, no irrational, no imaginary numbers. But if you are going to have carrier in high-tech you would have to work with imaginary numbers a lot.

Answer 8

Imaginary numbers emerged with 'numbers less than zero' which is needed to relate Cartesian co-ordinate system

A much earlier to that Vedic Mathematics does not relate numbers less than zero to computing!

It implies earlier "simpler number system" did work without numbers less than zero!

You may log on to for details of Vedic Mathematics manner to relate numbers!

Regards!

Answer 9

Electricity uses imaginary numbers. That's the only application I can think of.

Answer 10

Engineering and Scientific calculations

such as

Power Factor in Three-Phase Mains Power generation & distribution :-)