Why do we associate numbers with a line?

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Discussion Overview

The discussion centers around the conceptual and mathematical reasoning behind associating numbers with a number line. Participants explore the implications of zero's position on the line, the nature of positive and negative numbers, and the potential for alternative representations of numbers beyond the traditional line.

Discussion Character

  • Exploratory
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants question why numbers are placed on a number line and seek a deeper understanding of this convention.
  • There is a discussion about zero being the only point on the number line that is neither positive nor negative, with some suggesting it serves as a necessary junction for positive and negative numbers.
  • One participant proposes the idea of having two zeros on the number line, each serving as a junction for the smallest and largest positive and negative numbers.
  • Another participant mentions that numbers can also be associated with the complex plane, extending the concept beyond the number line.
  • Some participants assert that integers and positive integers are well-ordered, which contributes to their association with a line.
  • There is mention of the projective real line and the Riemann sphere as extensions of the number line, suggesting alternative geometrical representations of numbers.
  • A participant states that zero is larger than any negative number, which introduces a point of contention regarding the relative positions of numbers.

Areas of Agreement / Disagreement

Participants express various viewpoints regarding the association of numbers with a line, with no consensus reached on the necessity or implications of zero's position, the nature of the number line, or the potential for alternative representations.

Contextual Notes

Some discussions involve assumptions about the nature of infinity and the properties of numbers, which remain unresolved. The exploration of extensions to the number line introduces additional complexity that is not fully addressed.

GreenPrint
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Why do we associate numbers with a line?
It just dawned on me that I was never told why and that all the math I have studied so far assumed that numbers could be placed on a number line =/. Questioning this seems odd to me but I would like to exactly know why. I can't think of a reason.

Also the number zero... is the only point on the number line that is neither positive nor negative. Does it only exist because in order for us to place numbers on a number line we have to have some point to join negative numbers and positive numbers? Without it would be kind of hard to place both positive numbers and negative numbers on a line wouldn't? I've been thinking about it some and zero is... smaller than the most smallest negative number and smaller than the most smallest positive number... um also we assume that zero is indeed the only point on the number line were positive numbers and negative numbers meet and that there is no number which exists that is larger than the largest positive number +∞ and larger than the most largest negative number -∞? But yet we have zero as this point smaller than the smallest positive and negative numbers? Why can't we have both and have the number line behave like a line until you got to a point were numbers got so large that the number line bend and curved back to the negative numbers... or just have zero be the point joining -∞ and +∞ instead of it's location between the two smallest numbers and have a number line like this...?

what if there really were to zeros on the number line, one joining the smallest positive and negative numbers and one joining the largest positive and negative numbers... then when you divide by zero you can't because you are trying to divide by two different numbers or something...
 
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GreenPrint said:
Why do we associate numbers with a line?
It just dawned on me that I was never told why and that all the math I have studied so far assumed that numbers could be placed on a number line =/. Questioning this seems odd to me but I would like to exactly know why. I can't think of a reason.

Also the number zero... is the only point on the number line that is neither positive nor negative. Does it only exist because in order for us to place numbers on a number line we have to have some point to join negative numbers and positive numbers? Without it would be kind of hard to place both positive numbers and negative numbers on a line wouldn't? I've been thinking about it some and zero is... smaller than the most smallest negative number and smaller than the most smallest positive number... um also we assume that zero is indeed the only point on the number line were positive numbers and negative numbers meet and that there is no number which exists that is larger than the largest positive number +∞ and larger than the most largest negative number -∞? But yet we have zero as this point smaller than the smallest positive and negative numbers? Why can't we have both and have the number line behave like a line until you got to a point were numbers got so large that the number line bend and curved back to the negative numbers... or just have zero be the point joining -∞ and +∞ instead of it's location between the two smallest numbers and have a number line like this...?

what if there really were to zeros on the number line, one joining the smallest positive and negative numbers and one joining the largest positive and negative numbers... then when you divide by zero you can't because you are trying to divide by two different numbers or something...

We associate numbers with the plane, the complex plane.
 
3.14159253589 said:
We associate numbers with the plane, the complex plane.
We associate pairs of numbers with the plane.
 
3.14159253589 said:
We associate numbers with the plane, the complex plane.

That's an extension. The number line is just that - a line.

By the way, you missed a value of pi's decimal expansion in your name

3.141592653589
 
Zero is larger than any negative number.
 
Mentallic said:
That's an extension. The number line is just that - a line.

I think 3.14159253589 means that only real numbers are associated with a line?
 
Because they are ordered, and integers are "well ordered". You will learn more about ordering and largest this and smallest that in analysis.
 
ArcanaNoir said:
integers are "well ordered".

Positive integers are well-ordered.
 
  • #10
micromass said:
Positive integers are well-ordered.

Good call. Although they can be well-ordered.
 
  • #11
GreenPrint,
The reals are associated with a line because that is the simplest geometry that they can be embedded in.
An extension of the reals is the projective real line, which adds the value that you talk about, making the full set like a circle.
Extending that to complex numbers associates with a sphere, the Reimann sphere.

(I personally think you can take the extension further to define a filled in sphere -a ball- but this forum isn't for speculation).
 

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