# Explaining imaginary numbers to laypeople

1. Dec 19, 2015

### DaveC426913

I've had discussions with laypeople (of which, I am one) about real-world manifestations of imaginary numbers. We can never seem to find a satisfactory, concise example. I know they are used in real-world calculations for things like EM wavelengths in electronics, but if you aren't into electronics, that's not much use, especially if you have to go through the math just to get it.

I finally got my head around imaginary numbers when I discovered (and correct me if I'm wrong) that any calculation that can be done with imaginary numbers, can also be done without them; it's just arbitrarily more complicated.

So I've come up with what I think is an intuitive analogue of the practical applications of imaginary numbers. I hope to explain to the lay-person why we might use imaginary numbers, even where there's no direct, real-world manifestation of them.

I draw an analogy between imaginary numbers and virtual images.

Given a diagram of a flat mirror, a point light source and an observer (pic Q), construct the point of reflection and the angle between light and observer.
This can be done by determining the X and Y distances of both points and then calculating (or constructing) the angles, but it's a bit of work.

A far simpler method is to simply reflect the image into virtual space. The incident and reflected ray become trivial to construct (pic A). A simple straight line constructs both the path and angles.

I'm hoping to demonstrate that imaginary numbers do not have to have a real-world manifestation - just like the virtual light source and light ray do not have to exist in real space - in order to be useful in a real application.

Opinions? I know it's a bit of a stretch. Virtual images are not an example of imaginary numbers, merely an analogy.

I am open to a better way of intuitive, layperson explanations.

Last edited: Dec 19, 2015
2. Dec 19, 2015

### lychette

My experience in this area comes from trying to teach AC theory at 6th form (A level in england level)... a truly practical problem. I would arrange a practical set up of R and C in series...connected to a signal generator. I arrange frequency so that Vr = 3 volts and Vc = 4 volts.
You know the rest !!! the supply voltage is measured to be 5 volts. !! all of the pupils know that 3 + 4 = 5 is the perfect example of pythagoras theorem which means that the voltages need to be rerpresented on a graph. That leads me nicely into the need to represent numbers in a 'different way'

3. Dec 19, 2015

### DaveC426913

4. Dec 19, 2015

### lychette

I agree.... I teach physics so it is an easy and obvious way to get into it.
But in everyday life I like to ask the question (it is usually to physics students !!)... I walk 3 m then 4m ....how far away am I from the start?...it can be anything from 7m to (-)1m.
Once you do understand ideas it is difficult to realise how some people do not understand !!!..,.that is why we come here ?

5. Dec 19, 2015

### DaveC426913

But what does that have to do with imaginary numbers?

6. Dec 19, 2015

### lychette

to see how 3 + 4 can equal 5 requires an insight into showing numbers on something other than a straight line.
At this level I am reluctant to continue with any discussion

7. Dec 19, 2015

### DaveC426913

Ah, I see.

8. Dec 19, 2015

### lychette

youy already knew:).!!!!!!!... playing studid games....send your recycled electrons to some desevrving cause.
I get a feeling about what physics forum is about....may not stay too long

9. Dec 19, 2015

### DaveC426913

I think we're crossing signals here. I am not attempting to be mean or sarcastic or otherwise uncivil. I'm not sure what I said that makes you think I'm playing games, but my apologies. It would be a shame if you left thinking I, or PF, are not sincere and welcoming.

10. Dec 19, 2015

### Hornbein

Imaginary numbers are all about circles, spirals, and periodic phenomena like waves.

11. Dec 19, 2015

### mathman

Imaginary numbers are mathematical things, used by physicists and engineers. To understand the basic mathematics, start with square roots. If you confine yourself to real numbers, negative numbers don't have square roots. The imaginary numbers were invented (discovered?) to define square roots for negative numbers. Everything else follows.

12. Dec 19, 2015

### lychette

But you did already know what I said....you are not as innocent as you pretend

13. Dec 19, 2015

### lychette

But knowledge of electronics helps to understand something beyond electronics

14. Dec 19, 2015

### Staff: Mentor

3 + 4 cannot equal 5. It's possible for vectors whose magnitudes are 3 and 4 to add vectorially to one whose magnitude is 5, but that's different from saying "3 + 4 can equal 5."

Chill, dude! Dave's remarks in no way deserved your comments.

You could have made it clearer by stating that you were talking about vectors...

15. Dec 19, 2015

### lychette

This 'dude' was not aware that his contribution was about vectors.just pointing out to 'Dave' How I approached this topic in my teaching.
Dave deserves all the praise he gets
I am chilled

16. Dec 19, 2015

### FactChecker

I like to think of complex number system, with its definition of multiplication, as adding the fundamental operation of rotation in the complex plane. Vectors in R2 don't offer that as a basic addition/multiplication operation. Rotations are an important operation and the benefits of having it are profound. For instance, it gives easy geometric representations of the square root of -1 and of cyclic processes.

Yes they do. There may be other, more clumsy ways that avoid them, but complex numbers are the best way to describe cyclic behavior.

Last edited: Dec 20, 2015
17. Dec 19, 2015

### DaveC426913

I really, really don't understand. What did I know? What makes you think I knew it? And most of all, why do you think I would be acting with duplicity?

I'm reading the posts over and over. You mean how 3+4=5? I understand the logic there, yes. When I said 'Oh, I see' I was saying I understand why you are using that example to explain to your students.

Can we just start again?

18. Dec 20, 2015

### lychette

The OP relates to advice for 'lay people' it does not seem appropriate to me to attempt to explain imaginary numbers by introducing vectors just yet. Of course an understanding of vectors makes an understanding of imaginary numbers much easier !!
I like to use electrical analogies and it is very easy to make a series AC circuit of 2 components (C and R or R and L for example) where the 2 series voltages do not 'equal' (add up) to the supply voltage, hence 3+4 = 5 ( i would prefer to use the equivalent sign but I do not know how to get that!) or 1+1 = 1.4, or 6+7 = 9.2 etc
This shows students, who may not have met vectors yet, that the numbers do not simply add in the usual way. The 3+4 = 5 example quickly alerts them to pythagaros, and I take it from there...to representing the measured voltages graphically. then the process of learning and understanding goes forward.

19. Dec 20, 2015

### mathman

The idea of electrical circuits or light rays seems to be a very heavy handed way of explaining a purely mathematical idea. Think of the lay person as one who is a high school student in his/her freshman year, who would not have been exposed to any physics.

20. Dec 20, 2015

### lychette

But has been exposed to some maths, and knows about Pythagoras, and can use Pythagoras to add/ combine numbers in a particular way.