What is a negative number? intuitively

[reverse]

I can see how the idea of “nothing” or “0”, evolved from a hole or empty hand ..meaning that relative to objects (or contextual reality), there was a lack thereof.

 

But what on earth were they thinking when they went past nothing to create -1?

[Phi for All]

Perhaps it came about from the concept of borrowing. If you have 5 apples but borrow 10 to make pies, you have -5 apples.

 

This, of course, lead directly to deficit spending and politics.

[Mag]

Yeah, i was thinking the same thing.

 

And with your -5 apples, when you do get 5 apples, you will have none, because you have to give them to the guy you borrowed them from, or you could keep them... but then he'll sue you.

[Skye]

Nah, he lets you stay in debt and survive on apple pies so that you can buy his flint arrow heads.

[reverse]

lol...

 

logically then the inclusion of a - sign at the front, adds a wider understanding or relationship to the set of numbers.

 

I mean when you think of the number one...you never think...oh yea one more than nothing.

 

but when you add a "0" to the set...then you get that sort of relativity.

 

so....

 

logically....

 

when you add a "-1" to that set, you get a wider implied relativity.

 

but what kind of wider implied relativity...Huh?

[zaphod]

some people try to think too much. a negative number, intuitively, is a negative number.

[Tom Mattson]

I mean when you think of the number one...you never think...oh yea one more than nothing.

 

Sometimes I do.

 

when you add a "-1" to that set' date=' you get a wider implied relativity.

 

but what kind of wider implied relativity...Huh?[/quote']

 

Doesn't everyone who supports himself have an intuitive understanding of negative numbers? Every grown person is acutely aware of the consequences of spending more than he earns.

 

But if you want a variety of mathematical descriptions, the negative numbers are...

 

1. ...the objects that lie to the left of zero on the standard real line.

2. ...the additive inverses of the positive numbers.

3. ...what you obtain when subtracting two real numbers, the second of which is larger than the first.

[reverse]

Yes ..you are all quite correct

 

But I’m trying to figure this out more deeply.

 

For example …I just had one of those epiphany things where I totally understood that the number 1 and the number 9 are exactly the same just inverted in their displacement in the series.

 

Don’t ask.

 

now what is starting to bug me about numbers is the shifting frame of reference.

 

So for 1 to 9 we look at displacements of one …( but even this is not true if you get totally relative)

 

Then we add a 0.

 

Now we look at displacements of one relative to nothing. (the absolute context widens to include absence.)

 

Then we shift context again to include negative numbers…that almost implies a previous state now gone…like adding ”time” into the mix…

 

Why am I bugged…because if you started doing that

“man throws a ball from the front of a train to the back of a train, how fast is the ball going” stuff, you have to say the frame of reference for any answer to make sense.

 

I bet this all sounds strange.

[BigMoosie]

It makes logical sence to go into negative numbers. Before they were accepted as "numbers" the logic of being in dept of $5 still made sence, but to compare that $5 to another $5 that you made you need a way of differentiating the two. The way I see it negatives are just as invalid as positives, why if I have an apple do I say that I have +1 apple, it is just as valid to say that the rest of the world is -1 apple.

[the tree]

If x rocks were stolen from a pile of 30 then the cavemen would need to replace x rocks. If they know that 23 rocks are remaining then they'd need to take 23 from 30 to realise that they are missing 7 rocks. Apart from some painfully tedious trail and error, they wont get anywhere by addition and understanding subtraction is pretty much the same as understanding negative numbers.

[BigMoosie]

Hi Tree!

 

I dont think we are discussing the human brain's ability of grasping it but rather how our notation and general acceptance as the consept being a represented by a number. Or am I wrong?

[reverse]

Ok, I woke up this morning and my subconscious had been solving it overnight.

(don’t you love that) .

 

I'm not sure if you guys are going to be able to understand this.

 

a small shape is exactly the same as a large shape.

 

it just depends on how close you are standing to it.

 

so the number 1 is exactly the same as the number 2 ...it's just a matter of perspective.

 

I was thinking linearly before when I should have been thinking absolute three dimensionally and relatively.

 

get it?

[Tom Mattson]

I don't understand why people have such an urge to identify mathematical objects with physical objects, or to try to make mathematical arguments by analogy with physical arguments. It simply does not work.

 

Why is it so difficult to accept that mathematical objects are their definitions?

[Flareon]

Disregarding the whole minus apples and deficits and owing this or that, just imagine kinematics.

 

Okay, please bear with me, as this requires visualization.

 

If I were to mathematically describe my daily trip around town starting from the moment I were awake, my bedroom would be designated as the origin, or zero (and for simplicity's sake, let's keep this exercise 2D). To get to school, if I start walking to my car by facing east, I would count 1..2..3...meters. If I had decided to take the bus instead, I would walk to the stop by facing west, but would I count 1...2...3...meters again? Well, I could, but more accurate information would be given if I counted -1..-2...-3. Why? Because algebra tells us that now I am heading in the opposite direction of the car. Negative numbers tell us not only magnitude, but direction (confined to a single coordinate of course).

 

As we can agree, this is all arbitrary, and I can set the origin of zero at the nearby Starbucks, the White House, or Stone Henge. Then my trip would start not at zero, but at the distances from these places. This becomes moot however, when I would travel beyond and past any of these markers.

 

Here is my point: you need negative numbers to describe space. Forget the fact that negatives can be used to denote debt. Simply, positives indicate that the numbers point this-a-way, and the negatives indicate that the numbers point that-a-way. The reason we associate negatives with deficit is because we have long since set "having nothing" to zero, the origin.

[reverse]

Good point Tom.

 

I am aware of the tendency to do this.

 

Can I get back to you once I have thought about it.

 

I think it's got quite a lot to do with the way this allows you to solve one physical problem using the maths invented for a totally different application.

 

and that maths, truly must be another expression of the basic stuff of the universe, just like matter.

 

ie, they are the same thing.

[Flareon]

I don't understand why people have such an urge to identify mathematical objects with physical objects' date=' or to try to make mathematical arguments by analogy with physical arguments. It simply does not work.

 

Why is it so difficult to accept that mathematical objects [b']are[/b] their definitions?

 

I agree with you, except to add that I don't think it is as much as an urge, but a compulsion. Our brains are wired to use the concrete to understand the abstract, as one can argue that purely abstract thinking is next to impossible.

[reverse]

Disregarding the whole minus apples and deficits and owing this or that' date=' just imagine kinematics.

 

Okay, please bear with me, as this requires visualization.

 

.[/quote']

 

prefect.

 

now do that for a dolphin so you get another dimension (they move up and down as well)

 

then add a camera with a zoom lens (to allow for scale) ...and I think you have it!

[reverse]

French mathematician Blaise Pascal (1623-1662)

 

 

did not see the need for negative numbers.

 

 

 

Augustus de Morgan (1806-1871), an English mathematician,

thought numbers less than zero were unimaginable.

 

 

 

see:

http://www.und.edu/instruct/lgeller/negnum.html

[matt grime]

Here is my point: you need negative numbers to describe space. Forget the fact that negatives can be used to denote debt. Simply, positives indicate that the numbers point this-a-way, and the negatives indicate that the numbers point that-a-way. The reason we associate negatives with deficit is because we have long since set "having nothing" to zero, the origin.

 

We do not need negative numbers to describe space. We managed for many thousands of years with mathematics but without needing negative numbers whilst being perfectly able to navigate our way round. We do not *need* any mathematics. It is merely a useful tool. This whole "needing" physical interpretations held mathematics back for centuries. We are suffciently clever to be able to describe negative numbers as pairs of naturals, rationals as pairs of integers and complex numbers as pairs of reals, avoiding much ontological baggage in the process.

[zaphod]

i dont know if someone has already mentioned this, but after these threads about "what is this-or-that, intuitively?", here is the FINAL ANSWER to life, the universe, and everything:

 

it doesnt matter. mathematics is not about intuition. at all.

 

end of thread.

[reverse]

now listen here Beeblebrox,

 

think what you like, but math is a tool to do a job...

 

you may play with it if you wish...but that is not it's intention.

 

ask the dolphins.