Why Do Negative Numbers Exist?

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Discussion Overview

The discussion revolves around the existence and definition of negative numbers, exploring their purpose, representation, and implications in various mathematical contexts. Participants examine the conceptual underpinnings of negative numbers, their role in representing quantities such as debt, and the dimensionality of mathematical spaces.

Discussion Character

  • Exploratory
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question the necessity of negative numbers, suggesting that they do not exist in real life and proposing alternatives such as having more positive numbers.
  • Others argue that negative numbers are useful for representing concepts like debt and expenses, indicating that all numbers are abstract representations rather than physical entities.
  • One participant suggests that negative numbers serve to represent direction, while another points out that mathematical constructs like vectors and complex numbers can extend beyond one dimension.
  • There is a discussion about the dimensionality of vectors, with some asserting that vectors can represent any number of dimensions, while others express confusion about the limitations of dimensionality.
  • A later reply introduces the concept of n-dimensional Euclidean space, suggesting that negative numbers and zero are necessary for completing the group structure of natural numbers.
  • Another participant mentions that negative numbers are essential for closure under subtraction in the natural numbers, linking this to broader mathematical structures like rational and complex numbers.
  • Some participants highlight that negative numbers do appear in nature, particularly in contexts like debt or deceleration.

Areas of Agreement / Disagreement

Participants express a range of views on the existence and utility of negative numbers, with no clear consensus reached. Some agree on their mathematical necessity, while others challenge their relevance in real-life contexts.

Contextual Notes

Discussions include various assumptions about the nature of numbers and their representations, as well as unresolved questions regarding the dimensionality of mathematical constructs.

Who May Find This Useful

This discussion may be of interest to those exploring foundational concepts in mathematics, particularly in relation to number theory, dimensional analysis, and the philosophical implications of mathematical constructs.

tahayassen
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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?
 
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tahayassen said:
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.
 
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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?
 
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.
 
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?
 
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.
 
tahayassen said:
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.
 
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tahayassen said:
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?
 
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)
 
  • #10
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.
 
  • #11
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.
 
  • #12
When I think of negative numbers I think of debt and debt does exist as far as I know.
 

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