Why do we associate numbers with a line?

In summary, numbers are associated with a line because it is the simplest geometry to embed them in. The number zero exists as the point where positive and negative numbers meet. Having two zeros on the number line would cause issues with division. The extension to the complex plane allows for more complex numbers to be represented.
  • #1
GreenPrint
1,196
0
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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  • #3
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.
 
  • #4
3.14159253589 said:
We associate numbers with the plane, the complex plane.
We associate pairs of numbers with the plane.
 
  • #5
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
 
  • #6
Zero is larger than any negative number.
 
  • #7
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?
 
  • #8
Because they are ordered, and integers are "well ordered". You will learn more about ordering and largest this and smallest that in analysis.
 
  • #9
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).
 

1. Why do we use a line to represent numbers?

The use of a line to represent numbers is a visual representation of the numerical concept of order and magnitude. The line allows for easy comparison and sequencing of numbers, making it a useful tool for mathematical operations.

2. How did the concept of using a line to represent numbers originate?

The use of a line to represent numbers has been around for centuries, but it was popularized by the ancient Greek mathematician Euclid in his book "Elements". The concept of using a line to represent numbers was further developed by Rene Descartes in the 17th century, who introduced the idea of a number line.

3. Can a line accurately represent all numbers?

A line can only represent real numbers, which include both positive and negative numbers as well as fractions and decimals. Complex numbers and irrational numbers, such as pi, cannot be accurately represented on a number line.

4. How does the direction of the line affect the representation of numbers?

The direction of the line has no impact on the representation of numbers. A horizontal line can be used to represent positive numbers increasing from left to right, while a vertical line can represent positive numbers increasing from bottom to top. The direction is a matter of convention and does not change the numerical value of the line.

5. Are there other ways to represent numbers besides a line?

Yes, there are other ways to represent numbers such as using graphs, charts, and tables. These methods are often used to display large sets of data and allow for easier comparison and analysis. However, a number line remains a fundamental and commonly used tool for representing numbers in mathematics.

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