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

[jbriggs444]

So is the point that once we have chosen our origin (which we identify with the real number $0.0$), then the length of the interval between the origin and the point that we identify with the first integer, $1$, defines a length scale, which we refer to as being of "unit" length. Our metric is then defined in terms of this unit length such that the distance, $d(x,y)$ between any two real numbers is given by $$d(x,y)=\lvert x-y\rvert =\lvert y-x\rvert$$ which, in words, means that "the distance between $x$ and $y$ is $d(x,y)=\lvert x-y\rvert$ units", or, "there are $\lvert x-y\rvert$ intervals of unit length between $x$ and $y$".

Yes, that sounds reasonably accurate.

The "there are n intervals of unit length" justification that you give seems to makes an implicit assumption that you can move intervals around and compare or add up their lengths. That is an appealing assumption, but is not the only possibility. Non-linear metrics can be defined.

 

[Frank Castle]

Yes, that sounds reasonably accurate.

The "there are n intervals of unit length" justification that you give seems to makes an implicit assumption that you can move intervals around and compare or add up their lengths. That is an appealing assumption, but is not the only possibility. Non-linear metrics can be defined.

Would there be a better way to put it/ intuitively understand distances between real numbers?! In elementary introductions (that I have gone back and looked over), it often seems to be said that by choosing an origin and a unit length, defined as the length of the interval between 0 and 1, then one can use this to define the distance between any two real numbers via the Euclidean metric.

I'm really trying to justify in my mind this particular choice - is it necessary to define a unit length before defining the distance between any two real numbers?!

 

[jbriggs444]

Would there be a better way to put it/ intuitively understand distances between real numbers?! In elementary introductions (that I have gone back and looked over), it often seems to be said that by choosing an origin and a unit length, defined as the length of the interval between 0 and 1, then one can use this to define the distance between any two real numbers via the Euclidean metric.

I'm really trying to justify in my mind this particular choice - is it necessary to define a unit length before defining the distance between any two real numbers?!

One could equally well define the distance metric first and then verify the unit length along with any requirements for linearity, etc.

 

[Frank Castle]

One could equally well define the distance metric first and then verify the unit length along with any requirements for linearity, etc.

The problem I find intuitively with this way round, is what does, for example, $d(10,5)=5$ mean if we don't first define a unit of length? Naively, I've always thought of $d(10,5)=5$ as meaning that "$10$ is five units from $5$". Usually when we measure a distance it is relative to some unit of distance (physical examples would be 1 metre, or 1 inch, etc.)
If we defined the unit of length for real numbers to be something other than the length of the interval between 0 and 1, then the numerical value of $d(x,y)$ would change (although the length between $x$ and $y$ would not, of course).

 

[jbriggs444]

The problem I find intuitively with this way round, is what does, for example, $d(10,5)=5$ mean if we don't first define a unit of length?

Meaning? In mathematics? That's irrelevant. Things are what they are defined to be. That is as far as meaning goes.

We usually contrive to make definitions that formalize some pre-existing intuition, but that's irrelevant. Definitions are what they are, irrespective of what we think they might mean.

Edit: That did not come as a very friendly pronouncement. You are certainly right that in physics, we can attach meaning to distances. We do this with units of measure. We can lay out coordinate values on the number line in any number of ways. But if we do it with a linear scale, the only questions are where to place the origin and what scale factor to use. The size of the unit interval determines the scale factor.

 

[Frank Castle]

Meaning? In mathematics? That's irrelevant. Things are what they are defined to be. That is as far as meaning goes.

We usually contrive to make definitions that formalize some pre-existing intuition, but that's irrelevant. Definitions are what they are, irrespective of what we think they might mean.

Edit: That did not come as a very friendly pronouncement. You are certainly right that in physics, we can attach meaning to distances. We do this with units of measure. We can lay out coordinate values on the number line in any number of ways. But if we do it with a linear scale, the only questions are where to place the origin and what scale factor to use. The size of the unit interval determines the scale factor.

Fair enough. I just coming from a physics background I can't help myself but try to assign physical meaning to things!

In the case of the Euclidean metric, is it simply the case that we use such a linear scale, in the sense that the integers are equally spaced along the real number line, such that once we have chosen the origin then the size of the unit interval from $0$ to $1$ determines a measurement scale for the real number line.

Edit: @jbriggs444 sorry, just to check, is what I put above correct at all?

 

[Mark44]

Fair enough. I just coming from a physics background I can't help myself but try to assign physical meaning to things!

But that's not how things are generally done in mathematics.

In the case of the Euclidean metric, is it simply the case that we use such a linear scale, in the sense that the integers are equally spaced along the real number line, such that once we have chosen the origin then the size of the unit interval from $0$ to $1$ determines a measurement scale for the real number line.

The word "unit" is a backformation from "unity," which is the English translation of the Greek word "monas" (monad - a single unit; the number 1). So it's no coincidence that the interval from 0 to 1 is 1 unit.
It seems to me that you are making much more out of this than it really deserves.

 

[disregardthat]

Unit length is arbitrary, and while 1 usually is the most convenient choice, the unit length could be anything (though conventionally [0,1] is referred to as the unit interval). However, real numbers are fundamentally dimensionless. The unit length is given meaning once the real numbers are chosen to represent something (distance, speed, weight, temperature etc..), or once we decide on a unit abstractly, without referring to a precise type of measurement. Of course, one may freely translate between different units of measurement knowing their exact relation (for example kilometers to miles).

Take for example percentages. We say that 1% is the unit of percentages, while we simultaneously keep in mind that 1% is one part of a hundred, i.e. 1/100 or 0.01. In this context the real number 0.01 is the unit of measurement.

 

[Frank Castle]

But that's not how things are generally done in mathematics.

Yes, I understand that. It was more of a comment about my own brain being a bit resistant.

It seems to me that you are making much more out of this than it really deserves.

That's probably true.
My struggle really is, is the distance between real numbers, in terms of the Euclidean metric $\lvert x-y\rvert$, defined in terms of this unit length? If one chooses a length scale other than $1$ unit then I assume this would this affect the numerical value of the distance between numbers. For example, if we redefined the unit length to be the length of the interval between $0$ and $2$ and call it a "2unit", then the distance between $2$ and $7$ would be $3.5$ "2units", right?!
If not, then what does it mean to say that the distance between, for example, $3$ and $5$ is $2$? I've always taken this to mean that $5$ is $2$ units from $3$, and in general $x$ is $\lvert x-y\rvert$ units from $y$.
Maybe I'm just arguing semantics and looking for more meaning than there actually is, but I just want to make sure I've got the correct intuitive picture.

Sorry to go on, I know I'm probably being really stupid here - I've managed to confuse myself over something I thought I knew pretty solidly. (I think I've convinced myself that I don't understand it from delving more deeply into the maths of metric spaces, etc.)

 

[Mark44]

My struggle really is, is the distance between real numbers, in terms of the Euclidean metric $\lvert x-y\rvert$, defined in terms of this unit length?

Yes, of course, because the locations of x and y are defined in terms of this same length.

If one chooses a length scale other than $1$ unit then I assume this would this affect the numerical value of the distance between numbers. For example, if we redefined the unit length to be the length of the interval between $0$ and $2$ and call it a "2unit", then the distance between $2$ and $7$ would be $3.5$ "2units", right?!

Yes. If you use a different system of units for your length measurement, you're going to get a different result for the distance between two points. We deal with situations like this fairly often, especially when converting from one system of measure to another. For example, a league is an old nautical term for a distance equal to three nautical miles. An island that is 4 leagues offshore would be 12 nautical miles away.

Maps are commonly marked with scales in both miles and kilometers. The distance between two successive tickmarks on the kilometer scale (one km unit) is shorter than the corresponding distance between tickmarks on the mile scale.

If not, then what does it mean to say that the distance between, for example, $3$ and $5$ is $2$? I've always taken this to mean that $5$ is $2$ units from $3$

Why would you doubt this? How far apart are the two arrows in my drawing?

The most obvious answer is that they are two units of some kind apart, where the units are whatever distance each tickmark is from its neighbor. (In my crude drawing, the tickmarks are supposed to be equally spaced.)
In making this drawing, I am defining a "unit" by where I placed the tickmarks. This in turn defines the distance between the two points pointed to by the arrows. If I had made a finer division of this line segment by putting in more tickmarks, that would affect the locations of the points, and therefor, the distance between them.

, and in general $x$ is $\lvert x-y\rvert$ units from $y$.
Maybe I'm just arguing semantics and looking for more meaning than there actually is, but I just want to make sure I've got the correct intuitive picture.

As many times as you have asked this very same question in this thread, it doesn't seem that you are sure at all.

 

[Stephen Tashi]

Maybe I'm just arguing semantics and looking for more meaning than there actually is, but I just want to make sure I've got the correct intuitive picture.

The correct intuitive picture is to separate the concept of physical distance from the concept of distance between numbers. There is no specific physical distance between numbers. For example, a yard stick has one physical distance between "1" and "2" and meter stick has a different physical distance between them.

You could create a measuring stick where the distance between "0" and "2" was 1 inch and think of 1 inch as your "2unit".

In mathematics, the general term for the concept of distance is a "metric". You are correct that it is possible of define more than one metric on the set of real numbers. It's a cultural convention that when people mention the "distance" between two numbers, they have in the mind the particular metric given by $|x-y|$.

 

[Frank Castle]

Yes, of course, because the locations of x and y are defined in terms of this same length.

So is the point that we define the metric as $d(x,y)=\lvert x-y\rvert$ and then choose an origin, then one unit is simply $d(1,0)=1$. Given this, any real number $x$ is said to be a distance of $\lvert x\rvert$ units from $0$. The location of any real number $x$ is then the directed distance, i.e. either $+\lvert x\rvert$ (if $x>0$) or $-\lvert x\rvert$ (if $x<0$). Then, since (as you said) the location of each real number is defined in terms of this unit length, it is immediately obvious that the distance between any two numbers $x$ and $y$ is $\lvert x-y\rvert$ units.

Why would you doubt this? How far apart are the two arrows in my drawing?

I don't doubt this. I can visually see why this is the case, provided that the integers are arranged along the real number line such that there is an interval of unit length between each successive integer.

In making this drawing, I am defining a "unit" by where I placed the tickmarks. This in turn defines the distance between the two points pointed to by the arrows. If I had made a finer division of this line segment by putting in more tickmarks, that would affect the locations of the points, and therefor, the distance between them.

This is essentially what I've been trying to get at, in that is it the case that one has to define what a unit of length is, i.e. the length between consecutive integers in order for the the formula $\lvert x-y\rvert$ to "make sense". If one changed the unit of length, then the numerical value of $d(x,y)$ would change (as it would if one were making physical measurements in terms of meters or inches).

In mathematics, the general term for the concept of distance is a "metric". You are correct that it is possible of define more than one metric on the set of real numbers. It's a cultural convention that when people mention the "distance" between two numbers, they have in the mind the particular metric given by |x−y||x-y|.

When one defines this metric on the real numbers, is it implicitly assumed that the integers are equally spaced, and furthermore that there is a unit of length, defined as the length of the unit interval (or alternatively, the length of the interval between consecutive integers)?! Sorry I keep going on. Visually I can see that if one takes the distance between two integers on the real number line to be of unit length then one can then measure the distance between any two real numbers relative to this unit length (in the sense that $\lvert x-y\rvert$ means intuitively that $x$ is $\lvert x-y\rvert$ units from $y$).

 

[jbriggs444]

When one defines this metric on the real numbers, is it implicitly assumed that the integers are equally spaced

In general, there is no requirement that a metric preserve the property that the "distance" between consecutive integers is constant.

 

[Mark44]

So is the point that we define the metric as d(x,y)=∣x−y∣ and then choose an origin, then one unit is simply d(1,0)=1.

I think you have this backwards -- you choose the origin and the locations of 1, 2, etc., which defines the distance from 0 to 1 (and from 1 to 2 and so on).

 

[Frank Castle]

In general, there is no requirement that a metric preserve the property that the "distance" between consecutive integers is constant.

But, in the case of the real numbers, it is conventionally chosen that the metric does preserve the "distance" between consecutive integers, right?!

I think you have this backwards -- you choose the origin and the locations of 1, 2, etc., which defines the distance from 0 to 1 (and from 1 to 2 and so on).

So, by construction, one identifies the real numbers with points on a (geometric) line, choosing the location of the origin and the integers such that the they are equally spaced. We choose the metric such that the distance between any two real numbers is given by $\lvert x-y\rvert$, and given this, the distance between 0 and 1 is then taken to be the (base) unit of length, with the distance of each real number then being some multiple of this unit length away from the origin.
Would this be the correct chain of logic at all?

 

[Mark44]

In general, there is no requirement that a metric preserve the property that the "distance" between consecutive integers is constant.

As for example, a logarithmic scale or the scale used on some scales of a slide rule, such as the A and D scales in this image:

 

[Mark44]

But, in the case of the real numbers, it is conventionally chosen that the metric does preserve the "distance" between consecutive integers, right?!

Yes, that's the usual case.

So, by construction, one identifies the real numbers with points on a (geometric) line, choosing the location of the origin and the integers such that the they are equally spaced. We choose the metric such that the distance between any two real numbers is given by $\lvert x-y\rvert$, and given this, the distance between 0 and 1 is then taken to be the (base) unit of length, with the distance of each real number then being some multiple of this unit length away from the origin.
Would this be the correct chain of logic at all?

That's the way I look at it.

 

[Frank Castle]

Yes, that's the usual case.

I'm guessing this is why the Euclidean metric is chosen then, since it is always the case that any two consecutive integers are a unit distance apart ($\lvert (n+1)-n\rvert =1$ where $n$ is any integer)?!

That's the way I look at it.

Ok great, I think I'm finally "getting it" now. Sorry its been such a long process!

 

[jbriggs444]

I'm guessing this is why the Euclidean metric is chosen then, since it is always the case that any two consecutive integers are a unit distance apart ($\lvert (n+1)-n\rvert =1$ where $n$ is any integer)?!

Is that what you meant to write?

Edit: i.e. Did you mean to write that under the Euclidean metric is it always the case that ...

It is not always the case that any two consecutive integers are a unit distance apart. @Mark44 gave a nice graphical demonstration in #46.

 

[Frank Castle]

Is that what you meant to write?

I meant only in the case where we have chosen the integers to be equally spaced and defined distance via the Euclidean metric. If the distance between two consecutive integers were different one would use a different metric, right?

 

[jbriggs444]

I meant only in the case where we have chosen the integers to be equally spaced and defined distance via the Euclidean metric. If the distance between two consecutive integers were different one would use a different metric, right?

Yes, indeed.

 

[Frank Castle]

Yes, indeed.

Ok cool. Would you say (taking the last few posts) that I've finally understood things correctly? (I feel like I have now, but just want to clarify).

 

[jbriggs444]

Ok cool. Would you say (taking the last few posts) that I've finally understood things correctly? (I feel like I have now, but just want to clarify).

I think so, yes.

 

[Frank Castle]

I think so, yes.

Great, thanks for all you help (and patience)!

 

[Recycler]

This makes me wonder if a segment of number line actually needs a beginning and end point for each interval length.By definition,perhaps we cannot calculate its length unless we know the end points,but isn't the length the same whether the points are known or not?If n segments of length 1 are laid end to end,each having a start and end point,except for the last,which only has a start point,isn't the length the same as a line segment which has a defined endpoint?The only difference is one less defined point.In the line segment without a defined endpoint,doesn't the number of 1-length intervals equal the number of defined points?

 

[jbriggs444]

This makes me wonder if a segment of number line actually needs a beginning and end point for each interval length.By definition,perhaps we cannot calculate its length unless we know the end points,but isn't the length the same whether the points are known or not?If n segments of length 1 are laid end to end,each having a start and end point,except for the last,which only has a start point,isn't the length the same as a line segment which has a defined endpoint?The only difference is one less defined point.In the line segment without a defined endpoint,doesn't the number of 1-length intervals equal the number of defined points?

If you are considering the size of a set of real numbers, a relevant concept is "measure". The measure of an interval on the real number line is almost always taken to be equal to the difference in the endpoints. This applies whether the interval is open, closed or half-open. More generally, you can add or remove any countable number of points from a set without altering its measure.

One relevant notion of measure is https://en.wikipedia.org/wiki/Lebesgue_measure

As has been mentioned previously, measure (roughly "how big") and metric (roughly "how far") are separate concepts.

 

[Learner60]

Why is it that the distance between two real numbers $a$ and $b$ in an ordered interval of numbers, for example $a<x_{1}<\ldots <x_{n-1}<b$, is given by $$\lvert a-b\rvert$$ when there are in actual fact $$\lvert a-b\rvert +1$$ numbers within this range?!

Is it simply that, when measuring the distance between two real numbers we are counting the number of unit intervals that separate the two of them, and there will always be one less unit interval between the two numbers than the range of numbers between them?!

For example, suppose I have the ordered set of integers $(0,1,2,3,4,5)$, then the distance between 4 and 1 is of course $\lvert 4-1\rvert = \lvert 1-4\rvert = 3$, which is to say, there are 3 unit intervals between 1 and 4. Equivalently, one could arrive at this result by counting the number elements between 1 and 4, including the endpoint (4) but not the start point (1). However, if one includes both the start point and the endpoint then the number of elements between 1 and 4 is 4. Is the qualitative difference here that in the former case I am determining a relative quantity - the separation between 1 and 4, whereas in the latter case I am determining an absolute quantity- the number of elements ranging from 1 to 4?!Apologies if this is a really stupid question, but it's something that I've been thinking about recently, and how I would reason the answer.

Infinity?

 

[Mark44]

This makes me wonder if a segment of number line actually needs a beginning and end point for each interval length.By definition,perhaps we cannot calculate its length unless we know the end points,but isn't the length the same whether the points are known or not?If n segments of length 1 are laid end to end,each having a start and end point,except for the last,which only has a start point,isn't the length the same as a line segment which has a defined endpoint?The only difference is one less defined point.In the line segment without a defined endpoint,doesn't the number of 1-length intervals equal the number of defined points?

No, the endpoints of the segments don't have to be included.
The segments (0, 1), [0, 1), (0, 1], and [0, 1] all have exactly the same length -- one unit.
If you're not familiar with the parentheses/brackets interval notation, the same intervals are represented by these inequalities, respectively:
$0 < x < 1$ neither endpoint included
$0 \le x < 1$ left endpoint included
$0 < x \le 1$ right endpoint included
$0 \le x \le 1$ both endpoints included
Whether an endpoint of an interval is included or not makes not difference at all in its length.

 

[PeterDonis]

If one chooses a length scale other than $1$ unit

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.

 

[pwsnafu]

I'm guessing this is why the Euclidean metric is chosen then, since it is always the case that any two consecutive integers are a unit distance apart ($\lvert (n+1)-n\rvert =1$ where $n$ is any integer)?!

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 $d(x+1,x) = 1$.