Debt is a common way of visualizing it. Any sort of vector or relative scale can also be used. You might try using sea level and altitude, or as you mentioned temperature. Many things have no absolute scale, and only relative scales.

Negative Kelvin... I seem to recall reading about this as a possibility using lasers (around 1973+). It was counter-intuitive, of course, but there was a case to be made.

There is a concept in physiology that uses negative work...

Don't think these would help though.

Debt does seem the most appropriate given the examples all around us.

I agree with the previous two responses. Negative numbers came into being in Europe prominently due to their use in banks during the Renaissance era. Medici banks would double-record their accounts meaning a transaction would be recorded as a deduction from the first account and an addition into the second account. They represented the deduction using the red ink we're familiar with today and then eventually with the concept of negative numbers. That is probably the most natural way of explaining things because that was the way humans really first grasped the concept.

Negative kelvin is by definition impossible. 0 K represents absolute zero. Nothing can be below it.

You could introduce him to Cartesian coordinates and a simple graph with axes. At least that way you can ask him to place numbers on the x-axis himself and visualise the negative side.

I did this with my son at about the same age and he was fascinated. So much so that I then turned it into a Argand diagram with the real/imaginary axes and asked him to try to explain numbers that went up and down on the y-scale etc... The existence of the diagram seemed to help overcome the problem of reality: he just accepted that there was a whole world of stuff out there waiting to be explored.

flashman, that sounds wacky. For something to be below absolute zero it would have to maintain the possibility for negative motion on an absolute scale.

Negative work happens all the time, but it's on a relative scale. Friction is typically negative work, and gravity does negative work whenever you walk up stairs.

aum:

I realise that negative Kelvin goes against the definition but there was serious stuff written about it - that's why I can still remember it so many years on. Unfortunately, I cannot remember the references (negative memory).

Like Chrispminis said: debt. I now owe 5 apples. There were 3 apples, but someone apple dept collector took 8 of my apples away, now I owe him 5 because I didn't have enough apples to pay him off. I have -5 apples.

Honestly, with all the crazy things physicists are doing with quantum mechanics I wouldn't doubt that they somehow circumvented absolute zero...

Chrisp:

see above for negative K,

as for negative work - I seem to remember that the idea was concerned with leg muscles when a person climbed down from height.

flashmn see above for my second response. =D

Again, I can easily envision a situation where a muscle in the leg exerts a force upward, as a complementary muscle, when walking down stairs, in which it technically does negative work.

I'm in the North American auto manufacturing sector.

If your five year old ever gets this figured out, do you think that he could explain it to my accountants?

Bahahaha

@mapleleaf LOL.
Aren't they all too familiar with *negative* numbers

Take the kid to the sidewalk where the sidewalk is in blocks.
Take a piece of chalk.
Mark on a block start and then put the number +1 +2 +3 on the blocks that head towards your house.
Stand him at the start line and have him walk forward to the 1st 2nd and 3rd block.
Then put him back at the start line with you.
Tell him that he gets to ask you a really really hard question and if you get it wrong you get a penalty and have to go back one block each time.
What ever he asks you get it wrong and you then write down on the block -1 then -2 etc because that is how many blocks he has to get back to the start line.
They get the idea of a negative number means that it is further away from start ((0)) and that a positive number is going forward.
It has worked on two kids and one grandkid so far.

or just use a number line, and talk about negative numbers r less than zero. once u get into adding and subtracting, if its add, u face towadt the positives, minus towrards the negatives, and then if u add, u walk forward, and subtract u walk back. (so u r on 0, and its + (-4), u face backwards, but walk in the direction u r facing (backwards.)

thats not very clear, but once u explain it, it works.

Debt is probably the best way, as has been mentioned.

I'd give him 5 Popsicle sticks (or whatever) and tell him you want 8. You can then ask him how many short he is.

A good way to think about why absolute zero can't be crossed is that quantum particles "spin" depending on how much energy they have. You can't spin negatively, negative angular momentum is just spinning in the opposite direction

Not sure if telling your son he has -3 popsicle sticks will clarify things much :o

Negative absolute temperature can theoretically exist, but it doesn't mean what you think it means. In particular, it is actually HOTTER than hot. That's because, while you can't go below absolute zero, you can go above infinity and come around on the other side.

It all makes sense mathematically if you measure things using B = 1/T (where T is temperature measured in an absolute scale such as kelvins). Large values of B are cold, small values of B are hot, and T = 0 can never quite be reached (since division by zero is impossible, or if you prefer since B must stay finite). But negative values of T give negative values of B, which (being even smaller) describe even hotter temperatures. B = 0 is also possible, which is where T breaks down and shows that B really is the best mathematical description.

However, you can only actually get these negative absolute temperatures in weird quantum-mechanical situations, so you don't have to worry about them in real life. (Unless your real life involves solid-state engineering or something.)

As a mathematician, I think of signed numbers as giving potential changes in unsigned numbers. So +5 means to add 5 and -3 means to take away 3. I cannot actually have -3 apples, but it is possible for you to take away 3 of my apples (at least if I have enough). The original unsigned numbers become a special case of the signed numbers if you assume that you always start with 0 (since 0 + 5 = 5, for example).

I'm not saying that this is the best way to explain things to young children. But it's a little different than the other suggestions, so I mention it. (Although it's only a little distance from talking about changes in the amount of money that you have to talking about owing and being owed money.)

Negative work (in the physicists' sense of work) occurs whenever a force opposes a body's motion but the body still manages to move that way anyway (because of inertia or some other force).

So if I stand in front of a moving train, then when it hits me, I'm applying a small force to stop the train. But that is pretty small, and the train hardly slows down. I am having an affect on the train (however small), but as far as the quantity ‘work’ is concerned, I am doing negative work on it.

If Superman stands in front of the train instead, then even he is doing negative work on the train at first. But eventually the train slows down and stops, then starts moving backward. As soon as it starts moving in the same direction as Superman is pushing it, Superman is doing positive work on it.

Of course, the train is applying a force to me or Superman as well. In my case, the train is doing a positive amount of work on me, since I'm moving in the same direction as it's pushing me. On Superman, the train also does a positive amount of work at first, as it pushes Superman backwards. (Of course, Superman could stand still like a brick wall if he wanted to, but then he'd kill everybody on the train like a brick wall would, so he does have to move backwards for a bit as he slows the train down.) Then when Superman starts pushing the train backwards, the train is still pushing against Superman, but it's doing a negative amount of work since Superman is moving the other way.

You may notice that in all of these examples, the work done by the train on the person is opposite in sign to the work done by the person on the train. This will always happen if two objects are moving together. The forces that they produce on one another will be ‘equal and opposite reactions’ (that is the same absolute value but with opposite directions), while the distance moved will be the same (since they are moving together), and work is calculated as force times distance (with some business about vectors in there that don't matter to us since this is a one-dimensional example). So one will be positive and one will be negative, depending on which way the directions (of force and motion) match: positive for the same direction, but negative for opposite directions.

In this example, you can also understand work as a transfer of mechanical energy through direct contact; the train transfers energy to me, which sends me flying. Or it transfers energy to Superman until all of its kinetic energy is gone and it stops, after which Superman transfers energy to it as it starts moving again, only backwards. (Superman's motion is harder to calculate just in terms of kinetic energy, since he is also interacting with the ground and turning kinetic energy into heat through friction. So don't worry about that.)

Some would say you're using +ve and -ve numbers as a shortcut so you don't need to work with vectors (or that a -ve number is a +ve number with a -ve direction, making it a vector), so it'd still be a mathematical trick rather than an example of an actual negative quantity

Why not use the idea of debt as an example:
ie. Apples or candy sweets exchanged by each person and promising to owe back later;
this will allow you to introduce the negative by getting him to realise that next time he has aplles or candy he will be 'minus' the ones he owes to his friend.
It'll also introduce the idea of money and obligation and encourage him to see math as a way to do 'business'

The page on simple Wikipedia http://simple.wikipedia.org/wiki/Negative_number pretty much contains all examples already given here.

The idea of being the "opposite" might be not a bad way to explain it.

Toby Bartels, you have made my day: I just knew I'd read about negative absolute temperatures. Ha, forty years on and the old brain still does it...

EdiBirson wins with an outrageous lead over the geek squad. To teach a 5 year old - you have to think like a five year old. Or, better, know how five year olds think. A five year old can not use abstract reasoning - which is what everyone else was turning to. He has to understand it concretely - great used of the sidewalk. He has a concrete understanding. He has a process that manipulates tangible objects. Eventually, He can rehearse the process away from the tangeble manipulatives or structures by replaying the process mentally. Eventually he can abstract or generalize the process and then apply things like number lines, cartesean planes, credit/debit, opposite, etc. To start with the abstract will confuse and demoralize. Edi's process will have the kid doing the problem in his head in no time. I teach math.

Thank you for telling me I was wrong to show my son a graph: I shall now have to tell him that he didn't understand any of it after all...

Oh, and I used to teach maths (yep, there is an 's' where I come from).

I actually find number lines as a visual aid quite intuitive: I do not think we should be so quick to dismiss the abilities of the young.

Flashman, most 5 years don't get it. I am happy that your child is the exception, and it should be no suprise. Sorry, to have offended you, I can't say that I read every last post in this thread and evendently missed yours. Exceptional children in the hands of a patient tutor will excel. I will submit that, for the majority of people who don't teach math(s), when explaining these concepts to young children, the more concrete method would probably be advised.

By the way, stay off the coffee and switch back to tea.

Also, I have a number of university friends who were educated in England and the commonwealth. Some even taught maths for a time. I am well versed in the differnces between British and American English. Please don't fall into some bit of British sombery or pomposity just because I am an American and we've develop our own dialect. You'll start sounding like George W. or some other narrow minded American idiot you so readily believe populate this continent south of Canada.

remembar, this kid is 5, he wont be comprehending temps above infinity coming down to absolute 0

TheClark, I think you will find alot of exceptions to that rule indeed, particularly among children "fascinated about math", which is what this thread is about. I certainly know that I had understood two proofs of Pythagoras' at five. You surely have to teach those who find maths interesting and not so perplexing in a different way, which builds on the abstract nature of maths, otherwise can you really say that you are teaching then maths, rather than equipping them to learn it?

Also, you might do better than to label people who understand theoretical physics "the geek squad"

before you start explaining negative numbers, you should make sure the child understands the concept of "0".
As in:
5 + 0 = 5
anything + 0 = the same anything (x + 0 = 0 and 0 + x = 0)
and how to solve simple addition and subtraction problems.
2+3 =
5+4 =
5 - 2 =
then change the problems to something like this: (replace X with a box)
2 + X = 7
X - 5 = 4
3 - X = 0
once the child can solve those
give him/her something like this:
X + 3 = 0

My uncle once taught a five year old long division (6+ digit numbers divided by 2 digit numbers), so I don't suppose negative numbers should be too hard.

"He has to understand it concretely - great used of the sidewalk. He has a concrete understanding."

Great pun.

TheClark, it really depends on the child. The thread did get hijacked to talk about the possibilities of negative Kelvin and the existence of negative work as well as our economy... but Edi's idea was just a very specific way to demonstrate what was said earlier. I already said that any vector or relative scale can be used, and a Edi's sidewalk classically fits into that.

I can imagine some child pointing out that in Edi's example he's not negative distance from the start, he's just a distance in the opposite direction, there is no "real" negative distance except when we define a zero point and a direction along which we measure displacement. It still requires the child to think abstractly about the issue because they have to understand that there's a reference point and direction that's been defined rather arbitrarily. Concretely, a child would see only positive distance. Luckily, I think you underestimate the abstract thinking abilities of children.

I know that when I was younger, while I didn't really grasp literary metaphors, I was quick to grasp the abstraction inherent in mathematics and numbers.

Once he's grasped negative numbers here's a video I've made that explains Pythag Theorm. Enjoy x

http://www.youtube.com/watch?v=bP3i5dpezuk&feature=channel_page

For me "geek" is a term of endearment. This whole forum is a total geek out, please. I am a geek and really proud of it. I didn't realize that the denizens of forums that discuss Kirk vs Picard(Kirk by the way) or Dr. Who(Best Dr is Tom Baker) vs Star Trek were going to be upset with the used of that term in reference to them. Those are great threads by the way. It should be OK for geeks to use the "g" word.

@ thejoeman ‘remembar, this kid is 5, he wont be comprehending temps above infinity coming down to absolute 0’

No, you wouldn't want to use the discussion of negative absolute temperature to introduce negative numbers to children!

But if the kid is reading the thermometer, then I don't see what's wrong with using negative non-absolute temperatures to introduce negative numbers. Can something get colder than 0°C (or 0°F, whatever you use)? Sure, so what do we call that? Let's use negative numbers!

It's true that nothing can get colder than -273.15°C (or whatever it is in F), but that is not really important in everyday life. (Unless you're figuring out Boyle's Law or something.)

@ sceptic_ka:

Yes, I think you're right that you should talk about 0 before negative numbers. That is not so hard, because it is possible to have 0 apples (I have 0 apples right now!). It takes a bit of bending of the mind to realise that 0 (or 1, or even 2 sometimes) is a number, but once you get that, then you're ready to learn about other variations on the theme of number.

Okay, I am cool with all of the above.

Worth noting that the Greeks (Classical variety), amongst others, had no zero and avoided the concept.

@ flashman:
Yeah, poor saps. (Of course, they didn't have negative numbers either.)

you have to teach him the concept of 'direction' versus 'magnitude'