High School Why Is the Distance Between Two Real Numbers Given by Their Absolute Difference?

[Frank Castle]

You can't "choose a length scale" for the real numbers taken by themselves (at least not if you restrict yourself to the standard metric on them, which is what you appear to be doing).

What you appear to be imagining is something like this: I have two rulers, one twice as long as the other. I can call the length of the shorter ruler "1 unit" or the length of the longer ruler "1 unit"; switching from one convention to the other can be thought of as "choosing a length scale". But none of this changes anything about the real numbers. All it changes is the mapping between real numbers and lengths in the actual physical space in which the rulers exist--whether you assign the real number "1.0" to the length of the shorter ruler or the longer one.

When quantifying the distance between two real numbers though, isn't the standard approach to identify real numbers with points on a number line, choosing an origin and then identifying the other numbers such that the integers are equally spaced and then choosing a unit of distance to be the length of the interval between $0$ and $1$. With the choice of the Euclidean metric one then has that the distance between consecutive integers is always $1$ unit, a given real number $x$ is $\lvert x\rvert$ units from $0$, and the distance between any two real numbers $x$ and $y$ is $\lvert x-y\rvert$ units.

Err, no? At least that's not the only reason. Because we can define the metric $d(x,y) = 0$ if $x=y$ and $d(x,y)=1$ otherwise. Then $d(n, n+1) = 1$ for all integers. But d is not Euclidean metric.

What's special about Euclidean metric on $\mathbb{R}^1$ is that it is homogeneous ($d(\lambda x, \lambda y) = \lambda d(x,y)$ for all positive lambda) and translation invariant ($d(x+a,y+a) = d(x,y)$) and $d(x+1,x) = 1$.

Good point. I was a bit to loose with my statement there. Thanks for the details.

 

[Mark44]

Err, no? At least that's not the only reason. Because we can define the metric $d(x,y) = 0$ if $x=y$ and $d(x,y)=1$ otherwise. Then $d(n, n+1) = 1$ for all integers.

Is that what you meant to write? As opposed to d(m, n) = 1 for distinct integers m and n.

But d is not Euclidean metric.

What's special about Euclidean metric on $\mathbb{R}^1$ is that it is homogeneous ($d(\lambda x, \lambda y) = \lambda d(x,y)$ for all positive lambda) and translation invariant ($d(x+a,y+a) = d(x,y)$) and $d(x+1,x) = 1$.

 

[PeterDonis]

When quantifying the distance between two real numbers though, isn't the standard approach to identify real numbers with points on a number line, choosing an origin and then identifying the other numbers such that the integers are equally spaced and then choosing a unit of distance to be the length of the interval between $0$ and $1$.

No. The distance between two real numbers $x$ and $y$ is $| x - y |$. The two operations involved in this expression, subtraction and taking the absolute value, can be, and are, defined without reference to any of the things you mention.

Remember that this is a math forum, not a physics forum. Here the "real numbers" are not defined by any correspondence with lines or other objects that, at least in principle, have physical realizations. The real numbers are defined as a particular set with particular properties, which is constructed starting with the axioms of set theory. The same is true for properties of the real numbers like the Euclidean metric. So if you are trying to understand the real numbers as a mathematical object, you need to put aside any ideas you have about real numbers corresponding with lines, etc. Those ideas are completely irrelevant to the mathematical definition of real numbers and the Euclidean metric (or indeed any metric) on them.

An example of the kind of construction I am talking about is here:

http://www.math.wustl.edu/~kumar/courses/310-2009/peano.pdf

Notice that the construction starts with the natural numbers (1, 2, ...), which are constructed using the Peano axioms from set theory. From these the integers are constructed, then the rational numbers, then the real numbers. Nowhere is there any talk about a "number line" or any other such object. It's all set theory.

 

[PeterDonis]

Another quick reference is here:

http://math.mit.edu/classes/18.095/2015IAP/lecture1/padic.pdf

This focuses (the first part does, anyway) on constructing the reals from the rationals, but it has a good brief summary of the key sets at the start. It also discusses explicitly the absolute value function and how it is defined.

 

[Frank Castle]

No. The distance between two real numbers xx and yy is |x−y|| x - y |. The two operations involved in this expression, subtraction and taking the absolute value, can be, and are, defined without reference to any of the things you mention.

Remember that this is a math forum, not a physics forum. Here the "real numbers" are not defined by any correspondence with lines or other objects that, at least in principle, have physical realizations. The real numbers are defined as a particular set with particular properties, which is constructed starting with the axioms of set theory. The same is true for properties of the real numbers like the Euclidean metric. So if you are trying to understand the real numbers as a mathematical object, you need to put aside any ideas you have about real numbers corresponding with lines, etc. Those ideas are completely irrelevant to the mathematical definition of real numbers and the Euclidean metric (or indeed any metric) on them.

Why then, from an elementary perspective in mathematics, does one talk of a real number $x$ as being $\lvert x\rvert$ units from $0$ and more generally, two real numbers $x$ and $y$ being separated by $\lvert x-y\rvert$ units. Is it simply a heuristic device to enable one to, in a sense, visualise what is going on by giving a geometric interpretation to the distance between real numbers (on the so-called "real number line"). Or is it simply a matter of convenience to define a unit distance and then determine the distance between any two real numbers relative to this unit distance (i.e. as some sort of scaling of this unit distance)?!

Thanks for the links by the way, I shall have a read of them.

 

[PeroK]

Why then, from an elementary perspective in mathematics, does one talk of a real number $x$ as being $\lvert x\rvert$ units from $0$ and more generally, two real numbers $x$ and $y$ being separated by $\lvert x-y\rvert$ units. Is it simply a heuristic device to enable one to, in a sense, visualise what is going on, or is it simply a matter of convenience to define a unit distance and then determine the distance between any two real numbers relative to this unit distance?!

Thanks for the links by the way, I shall have a read of them.

The number 1 is sometimes called "unity". And the distance between 0 and 1 (using the normal definition of distance) is sometimes called a unit. But, as far as it goes, that's a mathematical definition of the word unit. It doesn't mean that there are any "units" involved in the sense that you use them in physics.

One interesting point is that some people like to say that the definite integral of a function is "units squared", but personally I never liked this. Mathematically a definite integral is itself a mapping from whatever you have (a function and an interval) to the set of Real numbers. It can be negative for example. It's not a mapping into a set of areas.

But, of course, you can interpret the definite integral as the area under a curve and if the function represents velocity against time, then the integral represents displacement. But, that's applying maths to a physical situation. Units such as $m, s, m/s$ are not inherenet in the mathematics.

 

[PeterDonis]

Why then, from an elementary perspective in mathematics, does one talk of a real number $x$ as being $\lvert x\rvert$ units from $0$

Please give some specific references. I think you are reading your own interpretation into sources that don't actually say this.

 

[Frank Castle]

The number 1 is sometimes called "unity". And the distance between 0 and 1 (using the normal definition of distance) is sometimes called a unit. But, as far as it goes, that's a mathematical definition of the word unit. It doesn't mean that there are any "units" involved in the sense that you use them in physics.

This is what I was alluding to. I get that the unit isn't referring to any actual physical unit of distance, but it seems to be the case, at least from a more elementary viewpoint, that distances between real numbers are expressed in terms of the distance between 0 and 1, i.e. in terms of units. For example, the distance of any real number $x$ relative to the origin is often referred to as being $\lvert x\rvert$ units

 

[Frank Castle]

Please give some specific references. I think you are reading your own interpretation into sources that don't actually say this.

Here are some examples:

http://eaton.math.rpi.edu/coursematerials/fall08/mk1500/AppdxE.pdf (on page 3)

http://pg.gda.pl/static/mathematics...asic_calculus/absolute/BC_t_absolutevalue.pdf

http://www.cengage.com/resource_uploads/downloads/113396074X_373363.pdf (on page 5)

http://www.sosmath.com/algebra/inequalities/ineq03/ineq03.html

http://www.mhhe.com/math/precalc/barnettpc5/graphics/barnett05pcfg/ch01/others/bpc5_ch01-04.pdf

 

[PeterDonis]

Here are some examples:

I see the word "units" used, and drawings using number lines, but I don't see anywhere that defines the real numbers in terms of "units" and number lines the way you have described in previous posts. So I don't think these references are examples of an "elementary perspective". They are just making use of a convenient expression and visualization when talking about derived concepts.

Note, for example, that in your third reference, top of page A2, it says:

Real numbers are represented graphically on the real number line.

It does not say that real numbers are defined in terms of the number line. The number line is a convenient graphical representation. That's all.

 

[Frank Castle]

I see the word "units" used, and drawings using number lines, but I don't see anywhere that defines the real numbers in terms of "units" and number lines the way you have described in previous posts. So I don't think these references are examples of an "elementary perspective". They are just making use of a convenient expression and visualization when talking about derived concepts.

Note, for example, that in your third reference, top of page A2, it says:
It does not say that real numbers are defined in terms of the number line. The number line is a convenient graphical representation. That's all.

I think the wires got a bit crossed in my previous posts, apologies for that.

I never meant that real numbers are defined in terms of the real number line (I realize that they are defined abstractly in terms of a set), merely that a one-to-one correspondence exists such that one can identify real numbers with points on a geometric line, the so-called "real number line". With this graphical representation, upon defining the distance between two real numbers as $d(x,y)=\lvert x-y\rvert$, we see from this graphical representation that any real number $x$ can be viewed as being a distance $d(x,0)=\lvert x\rvert$ from $0$. If one then one considers the unit length $d(1,0)=1$ as a "unit of length" along the real number line, then in this graphical representation, one can view a given real number $x$ as being a distance of $\lvert x\rvert$ units from the origin, and in general, any two real numbers $x$ and $y$, as being separated by a distance of $\lvert x-y\rvert$ units.

In the abstract, without using such a graphical representation, one then simply has that, for example, the distance between 8 and 3 is $d(8,3)=\lvert 8-3\rvert =3$ (with no units of any kind attached).

In the abstract sense, why is $d(x,y)$ referred to as the distance between two elements of a set? Is it because originally the notion was abstracted from the physical concept of distance between objects?

 

[PeterDonis]

In the abstract sense, why is $d(x,y)$ referred to as the distance between two elements of a set?

Because the function $d(x, y)$ defines a metric on the set (a metric is another abstract concept with an abstract definition independent of any graphical representation on a line, plane, etc.), and "distance" is the standard term for the number that is output by a metric $d(x, y)$ on a set when you plug in two elements $x$ and $y$ of the set.

Is it because originally the notion was abstracted from the physical concept of distance between objects?

Probably, but that's a question about history and terminology, not math.

 

[Frank Castle]

Because the function $d(x, y)$ defines a metric on the set (a metric is another abstract concept with an abstract definition independent of any graphical representation on a line, plane, etc.), and "distance" is the standard term for the number that is output by a metric $d(x, y)$ on a set when you plug in two elements $x$ and $y$ of the set.

Ok, fair enough.

Is what I wrote in the second and third paragraphs of my last post (#73) correct at all?

 

[Mark44]

In the abstract, without using such a graphical representation, one then simply has that, for example, the distance between 8 and 3 is $d(8,3)=\lvert 8-3\rvert =3$ (with no units of any kind attached).

Typo: d(8, 3) = |8 - 3| = 5, not 3

 

[jbriggs444]

Ok, fair enough.

Is what I wrote in the second and third paragraphs of my last post (#73) correct at all?

For me, the second paragraph in #71 does not say much of anything. It would help if you defined some terms so that one could distinguish between, for instance, "measured distance" between points on the geometric line, "defined distance" between real numbers as given by the metric and "derived distance" between points on the geometric line based on metric and the mapping you propose.

 

[PeterDonis]

Is what I wrote in the first paragraph of my last post (#73) correct at all?

Do you mean this?

With this graphical representation, upon defining the distance between two real numbers as $d(x,y)=\lvert x-y\rvert$, we see from this graphical representation that any real number $x$ can be viewed as being a distance $d(x,0)=\lvert x\rvert$ from $0$. If one then one considers the unit length $d(1,0)=1$ as a "unit of length" along the real number line, then in this graphical representation, one can view a given real number $x$ as being a distance of $\lvert x\rvert$ units from the origin, and in general, any two real numbers $x$ and $y$, as being separated by a distance of $\lvert x-y\rvert$ units.

I think the above still has issues. The main one is that none of what you say depends on the graphical representation; you're just restating what a "metric" or "distance function" is. A "unit" is just the result of applying the function $d(x, y)$ to the two members $x = 1$ and $y = 0$ of the set of real numbers. The statement $d(x, 0) = |x|$ is just a special case of the statement $d(x, y) = |x - y|$, with $y = 0$. All of this is true independently of any one-to-one correspondence between real numbers and points on a line.

 

[Frank Castle]

In the abstract, without using such a graphical representation, one then simply has that, for example, the distance between 8 and 3 is $d(8,3)=\lvert 8-3\rvert =3$ (with no units of any kind attached).
Typo: d(8, 3) = |8 - 3| = 5, not 3

Whoops, sorry. Yes, it should've been 5.

For me, the second paragraph in #71 does not say much of anything. It would help if you defined some terms so that one could distinguish between, for instance, "measured distance" between points on the geometric line, "defined distance" between real numbers as given by the metric and "derived distance" between points on the geometric line based on metric and the mapping you propose.

Ok.

Let $\mathbb{R}$ be the set of real numbers with the Euclidean metric defined on it, such that the distance between any two real numbers $x,y\;\in\mathbb{R}$ is $d(x,y)=\lvert x-y\rvert$.
If one the defines an injective map $S:\mathbb{R}\rightarrow E$, such that $S(x)=p=x$ (where $E$ is the one dimensional Euclidean space). We require that the mapping maps integers to equally spaced points in $E$ (I have to admit, I'm not sure how to do this?!), with the positive and negative integers equally spaced on either side of the chosen origin $\mathcal{O}\equiv 0$. Furthermore, we define the distance in $E$ by the same Euclidean metric: $d(p,q)=\lvert p-q\rvert$, where $p,q\;\in E$. As such, under the identity map, $S$ we have that $d(p,q)=d(S(x),S(y))=d(x,y)$.

Given this, the unit length in $E$ is defined as the distance of the interval between $0$ and $1$, $d(1,0)=1$, which we refer to as a "unit". As such, under this mapping, any real number $x$ can be interpreted as being a distance of $d(x,0)=\lvert x\rvert$ units from $0$, and indeed, the distance between any two real number, $x$ and $y$, as being $d(x,y)=\lvert x-y\rvert$ units.

Would this be any good?!

 

[Frank Castle]

Do you mean this?

Yes.

I think the above still has issues. The main one is that none of what you say depends on the graphical representation; you're just restating what a "metric" or "distance function" is. A "unit" is just the result of applying the function $d(x, y)$ to the two members $x = 1$ and $y = 0$ of the set of real numbers. The statement $d(x, 0) = |x|$ is just a special case of the statement $d(x, y) = |x - y|$, with $y = 0$. All of this is true independently of any one-to-one correspondence between real numbers and points on a line.

That is true.

I've tried to improve things in the above post (#77)...

 

[PeterDonis]

Let $\mathbb{R}$ be the set of real numbers with the Euclidean metric defined on it, such that the distance between any two real numbers $x,y\;\in\mathbb{R}$ is $d(x,y)=\lvert x-y\rvert$.

You have just defined the "one-dimensional Euclidean space".

If one the defines an injective map $S:\mathbb{R}\rightarrow E$, such that $S(x)=p=x$ (where $E$ is the one dimensional Euclidean space).

This is just the identity map, i.e., it is superfluous given what you've already defined. See above.

We require that the mapping maps integers to equally spaced points in $E$ (I have to admit, I'm not sure how to do this?!),

Your definition of $\mathbb{R}$ already does it; once you have the real numbers with the Euclidean metric, that automatically ensures that the distance $d(x, y)$ between any two consecutive integers $x$ and $y$ is the same.

 

[Frank Castle]

Your definition of $\mathbb{R}$ already does it; once you have the real numbers with the Euclidean metric, that automatically ensures that the distance $d(x, y)$ between any two consecutive integers $x$ and $y$ is the same.

Ah ok. Is this because the Euclidean metric is translation invariant?

What I'm really trying to justify is, as mentioned in a couple of the links that a put in post #69, why they refer to a number being $\lvert x\rvert$ units from $0$?!

 

[PeterDonis]

Is this because the Euclidean metric is translation invariant?

I suppose that would be one way of looking at it, yes. But the fact that $| x - y |$ is the same for any two consecutive integers $x$ and $y$ is provable from the set theoretic construction of the integers and the definition of subtraction and absolute value, plus the definition of what "consecutive" integers are. (And all of the properties of the integers, including this one, are preserved once we construct the reals and view the integers as a subset of the reals.)

What I'm really trying to justify is, as mentioned in a couple of the links that a put in post #69, why they refer to a number being $\lvert x\rvert$ units from $0$?!

At this point I'm not sure what you would consider a justification. To me it's just a matter of how you want to define the term "unit"; it has nothing to do with the actual math, it's just a matter of how you want to use ordinary language.

 

[Frank Castle]

I suppose that would be one way of looking at it, yes. But the fact that $| x - y |$ is the same for any two consecutive integers $x$ and $y$ is provable from the set theoretic construction of the integers and the definition of subtraction and absolute value, plus the definition of what "consecutive" integers are. (And all of the properties of the integers, including this one, are preserved once we construct the reals and view the integers as a subset of the reals.)

How straightforward is it to prove these properties? (Do you know of a good set of notes that discusses this in detail?)
Also, how does one define what "consecutive" integers are, does one define a function that recursively maps from $0$ to consecutive integers?!

At this point I'm not sure what you would consider a justification. To me it's just a matter of how you want to define the term "unit"; it has nothing to do with the actual math, it's just a matter of how you want to use ordinary language.

Ah ok. So it is then simply a matter of choice that one refers to the distance between $0$ and $1$ as a "unit", and then since all integers are equally spaced (when the Euclidean metric has been chosen), one can say that any integer $n$ is "$n$ units" from $0$ and then, for any real number, one can think of it as being "$\lvert x\rvert$ units" from $0$, or two real numbers being "$\lvert x-y\rvert$ units" apart?! (I guess this can be an appealing way to look at things when one uses the geometric notion of a number line, since it acts as a kind of basis to compare distanc)

 

[PeterDonis]

how does one define what "consecutive" integers are

Two distinct integers $x$ and $y$ are consecutive if there is no integer $z$ between them, i.e., no $z$ such that $x < z < y$ or $y < z < x$.

How straightforward is it to prove these properties?

The property to be proven is that, if two integers $x$ and $y$ are consecutive, as defined above, then $| x - y | = 1$. Here is a sketch of a proof:

(1) Since for any $x$ and $y$, we have $| x - y | = | y - x|$, we can restrict attention to the case $y < x$, so $| x - y | = x - y$ (i.e., we can drop the absolute value operation).

(2) For the case $x = 1$, $y = 0$, we have $x - y = 1 - 0 = 1$.

(3) Any pair of consecutive integers $x$ and $y$ with $y < x$ can be expressed as $x = 1 + w$, $y = 0 + w$, where $w$ is an integer. (Can you see why?)

(4) Thus, for any pair of consecutive integers $x$ and $y$, we have $x - y = (1 + w) - (0 + w) = 1 - 0 = 1$.

 

[PeterDonis]

So it is then simply a matter of choice that one refers to the distance between $0$ and $1$ as a "unit"

Yes.

 

[Frank Castle]

(Can you see why?)

Is this simply because $0+w=w< w+1$ and there is no integer $z$ such that $w<z<w+1$ since $w<w+1\Rightarrow 0<1$ and there is no integer between $0$ and $1$?!

Yes.

Is the reasoning a gave for this choice correct - by choosing this "basis length" one can then refer to the distance between and two real numbers in terms of this unit length (since any two integers will be a "unit" distance apart, and any other real number will be inbetween two integers and so will be some amount of "units" distance from $0$, or any other real number). However, it is important to bear in mind that it is not a "unit of distance" as in the physical sense, it is simply a "book-keeping" device to enable one to intuitively think of distances between real numbers as lengths along a number line. In general, for example, the distance between $6$ and $1.5$ is $4.5$ and one does not need to refer to a unit length (as in "$4.5$" units), it is simply $d(6,4.5)=4.5$, nothing more, nothing less?!

 

[PeterDonis]

Is this simply because $0+w=w< w+1$ and there is no integer $z$ such that $w<z<w+1$ since $w<w+1⇒0<1$ and there is no integer between $0$ and $1$?!

Basically, yes.

Is the reasoning a gave for this choice correct

To me it doesn't look like "reasoning"; it looks like you're just restating the same thing over and over again in different words.

by choosing this "basis length" one can then refer to the distance between and two real numbers in terms of this unit length

There is no "choosing the basis length" as a separate step. The "basis length" is just $d(0, 1)$; as I said before, it's just a special case of $d(x, y)$ with $x = 0$ and $y = 1$. There is no need to define that special case in order to give meaning to $d(x, y)$ in general; in fact it's the other way around, you define $d(x, y)$ in general and then, if you must, use the word "unit" to refer to $d(0, 1)$.

 

[Frank Castle]

Basically, yes.

Would there be a better way to explain why?

To me it doesn't look like "reasoning"; it looks like you're just restating the same thing over and over again in different words.

You're right, I think I'm massively over thinking it on this point. I just wanted to clarify that it is a valid choice (and why one might choose it, as some of the notes in the links a gave have done)?!

 

[PeterDonis]

Would there be a better way to explain why?

Only if you want a mathematician's level of rigor.

 

[Frank Castle]

Only if you want a mathematician's level of rigor.

I'd be keen to see it if that's ok?!

One last thing about the unit length $d(1,0)$. If one chooses to refer to this as a "unit", then is the reason why one can then speak of a given real number $x$ being $\lvert x\rvert$ "units" from $0$ because one can write $d(x,0)=\lvert x\rvert d(1,0)=\lvert x\rvert\text{ ''units"}$, or is it simply by noting that there is a "unit" distance between each consecutive integer, and so for integers $n$, it must be that if $1$ is a "unit" from $0$, then $n$ is $d(n,0)=\lvert n\rvert$ "units" from $n$ (since there are $n$ consecutive integers between $0$ and $n$, each separated by a "unit"). Then, more generally, for any real number we have that it will be somewhere within the interval between two consecutive integers so it will always be some multiple of "units" away from $0$ (for example, $1.17\in\left[1,2\right]$ and as such is a distance $d(1.17,0)=1.17$ "units" from zero).

 

[PeterDonis]

I'd be keen to see it if that's ok?!

I'm not a mathematician so I won't try to give that level of rigor. You would probably need to consult a textbook on analysis.

is the reason why one can then speak of a given real number $x$ being $\lvert x\rvert$ "units" from $0$...

You're continuing to overthink this. "Unit" is just a word. We choose to apply it to a certain number, the number obtained by evaluating $d(1, 0)$, the metric $d$ applied to the real numbers $1$ and $0$. Once we have made that choice, we can observe that all other numbers $d(x, y)$ bear a certain relationship to the number $d(1, 0)$ that we are calling a "unit". That's all there is to it. There isn't any more. All you're doing is continuing to say the same thing over and over again in different words. You aren't adding any new understanding.

I strongly suggest that you take a step back and think about the above before posting again on this subject. At this point I'm going to close this thread since it has clearly run its course. If you have a genuinely new question after thinking things over, you can start a new thread.