Explaining imaginary numbers to laypeople

[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.

 

[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 V_r = 3 volts and V_c = 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'

 

[DaveC426913]

...but if you aren't into electronics, that's not much use...

 

[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 realize how some people do not understand !..,.that is why we come here ?

 

[DaveC426913]

.. I walk 3 m then 4m ...how far away am I from the start?...it can be anything from 7m to (-)1m.

But what does that have to do with imaginary numbers?

 

[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

 

[DaveC426913]

to see how 3 + 4 can equal 5 requires an insight into showing numbers on something other than a straight line

Ah, I see.

 

[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

 

[DaveC426913]

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

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.

 

[Hornbein]

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

 

[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.

 

[lychette]

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.

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

 

[lychette]

But knowledge of electronics helps to understand something beyond electronics

 

[Mark44]

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

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."

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

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

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

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

 

[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

 

[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.

I'm hoping to demonstrate that imaginary numbers do not have to have a real-world manifestation

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

 

[DaveC426913]

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

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?

 

[lychette]

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...

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.

 

[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.

 

[lychette]

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

 

[Mark44]

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."

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

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.

If an instructor is going to teach students about imaginary numbers, it seems to me that the most natural concept to lead up to complex numbers is a short presentation of vector addition. I really hope you aren't teaching your students that $3 + 4 = 5$ (or even that $3 + 4 \equiv 5$), or $1 + 1 = \sqrt{2}$, none of which is true, . It would be a lot less confusing for them to say that you are working with Pythagoras' Theorem and how vectors add. Presumably the students know a bit of right triangle trig, and are able to determine the hypotenuse when the lengths of the two sides are 3 and 4, or when the two sides are both 1.

In either case, 3 + 4 is still 7 and 1 + 1 is still 2.

 

[DaveC426913]

... the most natural concept to lead up to complex numbers is a short presentation of vector addition.

Can you give an example of how vector addition leads to imaginary numbers?

I mean, I know you can plot reals on the x-axis and imaginary on the y-axis and then operate on it like it's Cartesian, but...

 

[Mark44]

Can you give an example of how vector addition leads to imaginary numbers?

Vector addition doesn't lead to imaginary numbers -- the two systems, vectors in the plane, and complex numbers, are in many ways analogous.

I mean, I know you can plot reals on the x-axis and imaginary on the y-axis and then operate on it like it's Cartesian, but...

Consider the vectors $\vec{u}$ = <1, 2> = 1i + 2j, and $\vec{v}$ = <2, 2> = 2i + 2j, where i is a unit vector in the horizontal direction (but not the imaginary number i), and j is a unit vector in the vertical direction. $\vec{u} + \vec{v}$ = <3, 4> = 3i + 4j.

Vectors add component-wise, exactly the same as do complex numbers. If $z_1 = 1 + 2i$, and $z_2 = 2 + 2i$, then $z_1 + z_2 = (1 + 2) + (2 + 2)i = 3 + 4i.$ Here, i is the imaginary unit, $\sqrt{-1}$.

In the vector example, $|\vec{u} + \vec{v}| = \sqrt{3^2 + 4^2} = 5$. In the complex example, $|z_1 + z_2| = \sqrt{3^2 + 4^2} = 5$.

 

[pbuk]

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 don't think you will find a real-world manifestation of an imaginary number.

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.

I don't think so - how about x² = -1 ?

So I've come up with what I think is an intuitive analogue of the practical applications of imaginary numbers...
Opinions?

I don't like the analogy because it confuses the technical meaning of "imaginary" in "imaginary number" with its everyday meaning. You see to a mathematician, the number i which is a solution to i² = -1 is no more imaginary than the number 3 which is the solution to x = 2 + 1: both are constructed from axioms and definitions that are equally arbitrary.

This is the difference between mathematics and physics: in physics we observe things about the real world, construct theories which explain the observations and conduct experiments to test the theories. In mathematics we construct axioms "out of thin air" and use these axioms to construct statements which we show to be consistent which we then call theorems and say that they are true (if we come up with an inconsistency we throw one or more of the axioms away and start again). There are cross-overs between the two disciplines - for instance Euclid observed that if you have a line and a point not on that line you can construct exactly one line that passes through the point that does not cross the original line and from this constructed what we still call Euclidean geometry. However later mathematicians realized that Euclid's observation of parallel lines in the real world was not necessary to construct a consistent geometry but (apart from the special case of mapping the Earth's surface) this was not thought to have any relevance to the real world; later still general relativity showed that in fact the real world IS non-Euclidean... see your other current thread!

 

[lychette]

If an instructor is going to teach students about imaginary numbers, it seems to me that the most natural concept to lead up to complex numbers is a short presentation of vector addition. I really hope you aren't teaching your students that $3 + 4 = 5$ (or even that $3 + 4 \equiv 5$), or $1 + 1 = \sqrt{2}$, none of which is true, . It would be a lot less confusing for them to say that you are working with Pythagoras' Theorem and how vectors add. Presumably the students know a bit of right triangle trig, and are able to determine the hypotenuse when the lengths of the two sides are 3 and 4, or when the two sides are both 1.

In either case, 3 + 4 is still 7 and 1 + 1 is still 2.

T

First of all, I am not an 'instructor', I am a teacher and have been for the best part of 40 years. I know what has worked for me and I do not think that these discussions should be places where 'mentors' Should offer advice on how to teach (not instruct!).
It is a distraction from seeking alternative ways of 'teaching' , who is to judge the value of alternative approaches?
I have been here for a very short time and I am disillusioned by responses, especially from mentors.
I feel that these forums will be better off without my input.

 

[DaveC426913]

I have been here for a very short time and I am disillusioned by responses, especially from mentors.
I feel that these forums will be better off without my input.

Lychette, I think you are taking unwarranted umbrage.

Everyone is entitled to contribute. Mark 44 is has his opinion about teaching methods (a door you opened yourself); he is not criticizing you.

He gets to offer his opinion. We all get to.

 

[S.G. Janssens]

I feel that these forums will be better off without my input.

Not necessarily, but probably this thread would be better off when your input is about complex numbers proper. Let's discuss mathematics.

 

[Mark44]

If an instructor is going to teach students about imaginary numbers, it seems to me that the most natural concept to lead up to complex numbers is a short presentation of vector addition. I really hope you aren't teaching your students that $3 + 4 = 5$ (or even that $3 + 4 \equiv 5$), or $1 + 1 = \sqrt{2}$, none of which is true, . It would be a lot less confusing for them to say that you are working with Pythagoras' Theorem and how vectors add. Presumably the students know a bit of right triangle trig, and are able to determine the hypotenuse when the lengths of the two sides are 3 and 4, or when the two sides are both 1.

In either case, 3 + 4 is still 7 and 1 + 1 is still 2.

T
First of all, I am not an 'instructor', I am a teacher and have been for the best part of 40 years. I know what has worked for me and I do not think that these discussions should be places where 'mentors' Should offer advice on how to teach (not instruct!).
It is a distraction from seeking alternative ways of 'teaching' , who is to judge the value of alternative approaches?
I have been here for a very short time and I am disillusioned by responses, especially from mentors.
I feel that these forums will be better off without my input.

I was a teacher/instructor for 21 years, including 18 years teaching mathematics at a community college in the Seattle area. IMO, this gives me the right to evaluate and critique different teaching methods. An important criterion for any teaching method is that it not promote stuff that isn't true, such as 3 + 4 = 5.

 

[LCKurtz]

@lychette: You might find my article "There's nothing imaginary about complex numbers" interesting as a way to broach the subject to beginners. It's at https://math.la.asu.edu/~kurtz/complex.html

 

[lychette]

Thank you for this LC...I am very familiar with the content of your article. it is standard textbook analysis.
I do not want to labour the point, but my need to introduce complex numbers has been in practical physics...eg: a simple series circuit consisting of a power supply and 2 components in series. In AC circuits the measured supply voltage does not equal the sum of the voltages across the series elements (using standard digital or analogue meters). In fact I ARRANGE the voltages to be 3V, 4V and 5V. It does not take long for students to realize that these values do not add in an arithmetic way, and it is usually a pleasant learning experience that these numbers must be represented vectorially. The door is open from this point.