Why Do Negative Numbers Exist?

[tahayassen]

Why did we define negative numbers? They don't exist in real life. Was their purpose to add negative numbers instead of subtracting positive numbers? Why can't we have more positive numbers that are twice as positive as regular positive numbers? Or numbers that are twice as negative as regular negative numbers?

 

[HallsofIvy]

Why did we define negative numbers? They don't exist in real life.

Then you must have a very limited real life! Well, in fact, NO numbers "exist" in real life. We use numbers to represent things in real life. And there are many reasons why we would want to represent some things by negative numbers. For example, if we represent money coming in (income) by positive numbers it makes sense to use negative numbers to represent money we have to pay.

Was their purpose to add negative numbers instead of subtracting positive numbers? Why can't we have more positive numbers that are twice as positive as regular positive numbers? Or numbers that are twice as negative as regular negative numbers?

What do you mean by "twice as positive"? I would say that if "x" is a positive number then "2x" is "twice as positive". But 2x, of course, already exists.

 

[tahayassen]

Okay, so negative numbers are there to represent direction. Why limit our direction in one dimension?

If positive goes to the right, and negative goes to the left, what about numbers that go up or down?

 

[HallsofIvy]

Well, problems that involve negative numbers are typically "one dimesional".
But we don't limit direction. There are many ways to include two dimensions, using vectors or complex numbers.

 

[tahayassen]

Vectors only go up to two dimensions though. What about the third dimension and so on? Wouldn't we able to do this forever? Why limit ourselves to a certain number of dimensions?

 

[conquest]

Well we do have n-dimensional eucilidian space so that takes care of that direction problem!

negative numbers and 0 in hindsight are neccesary to complete the group structure on the natural numbers. Which means we have some kind of multiplication map on the naturel numbers (i.e. addition). This is associative so a+(b+c)=(a+b)+c, where a, b and c are natural numbers. If you put 0 in there you have a unit element which means for any natural number a we have a+0=0+a=a (it's nilpotent). but then we also need an inverse so for every natural number a we want a natural number a^-1 such that a + a^-1 =0 and there inverses are exactly the negative numbers. A group structure on the natural numbers with addition (which is also quite natural) thus means we need 0 and negative numbers.

Also negative numbers certainly do show up in nature. Just think about debt or decceleration.

 

[D H]

Why did we define negative numbers?

So we could go into debt, Polonius' advice to Laertes notwithstanding. And I just can't resist myself,


Getting serious now, negative numbers make the natural numbers closed under subtraction. Similarly, the rationals are the closure of the integers under division, the reals are the closure of the rationals via limits, and the complex numbers are the algebraic closure of the reals.

 

[Char. Limit]

Vectors only go up to two dimensions though. What about the third dimension and so on? Wouldn't we able to do this forever? Why limit ourselves to a certain number of dimensions?

This is absolutely not true. A vector is an n-dimensional representation of a number, for any whole number n. 2-vectors represent two dimensions, 3-vectors represent 3 dimensions, so on and so forth. And if that doesn't satisfy you, we also have tensors, and I actually have no idea what they represent. Just curious, have you gone into complex numbers as of yet?

 

[conquest]

Tensors represent linear maps from vector spaces to the real numbers. The space of all tensors is in turn again a vector space with dimension the product of the dimensions of the vector spaces on which the elements are linear maps.

so the space of tensors from 3 copies of 3 dimensional real space has dimension 27
(but since 3 copies of 3 dimensional real space is isomorphic to 9 dimensional space this is just the space of 9x9 matrices which indeed has dimension 27)

 

[mathgenie99]

Hi friends,
I am Mohit S.Jain. I am new here. I want to share my ideas with others.
I am glad to join this forum.

 

[mathgenie99]

Hi friends,
I am Mohit S.Jain. I am new here. I want to share my ideas with others.
I am glad to join this forum.

 

[Containment]

When I think of negative numbers I think of debt and debt does exist as far as I know.