Distance between real numbers

[Frank Castle]

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.

 

[TeethWhitener]

Is the distance between 1 and 4 greater than, less than, or equal to the distance between 1.5 and 4.5? Why?

 

[Frank Castle]

Is the distance between 1 and 4 greater than, less than, or equal to the distance between 1.5 and 4.5? Why?

It is equal to, since there are still 3 unit intervals between 1.5 and 4.5. However, there are more than 4 numbers between 1.5 and 4.5...

 

[TeethWhitener]

However, there are more than 4 numbers between 1.5 and 4.5...

How many numbers are you counting between 1.5 and 4.5?

 

[Frank Castle]

How many numbers are you counting between 1.5 and 4.5?

Well, if we are counting in halves, i.e. 1.5, 2.0, 2.5,... then there are 6 numbers.

Is the point that we can arbitrarily partition up each unit interval into as many numbers as we please and still end up with the same number of unit intervals between two numbers so the distance between two numbers is something that is intrinsic as opposed to the arbitrary partitioning of the interval between them?!

 

[TeethWhitener]

Is the point that we can arbitrarily partition up each unit interval into as many numbers as we please and still end up with the same number of unit intervals between two numbers so the distance between two numbers is something that is intrinsic as opposed to the arbitrary partitioning of the interval between them?!

Yes. Well, kind of. There are three answers to your question. The first is that we simply define the distance between $a$ and $b$ to be $|a-b|$in $\mathbb{R}$. The second, more intuitive, answer is that the number of real numbers between $a$ and $b$, where $a \neq b$, is uncountable, so the notion of counting the numbers between $a$ and $b$ to determine a distance doesn't make any sense in $\mathbb{R}$. The third, most intuitive, answer is this: say we live at mile marker 1 and the grocery store is at mile marker 4. If the Department of Transportation comes along and relabels our mile marker as 1.5 and the grocery store's as 4.5, the simple act of relabeling the mile markers shouldn't change the distance between our house and the grocery store, should it?

 

[Frank Castle]

Yes. Well, kind of. There are three answers to your question. The first is that we simply define the distance between $a$ and $b$ to be $|a-b|$in $\mathbb{R}$. The second, more intuitive, answer is that the number of real numbers between $a$ and $b$, where $a \neq b$, is uncountable, so the notion of counting the numbers between $a$ and $b$ to determine a distance doesn't make any sense in $\mathbb{R}$. The third, most intuitive, answer is this: say we live at mile marker 1 and the grocery store is at mile marker 4. If the Department of Transportation comes along and relabels our mile marker as 1.5 and the grocery store's as 4.5, the simple act of relabeling the mile markers shouldn't change the distance between our house and the grocery store, should it?

So, is the intuitive answer simply that the distance between two points along the real number line should not depend on how we separate up the distance between them - it is an intrinsic property. It is not well-defined to determine the distance between two real numbers in terms of counting the numbers between them, since the answer would depend on how we partition up the interval, e.g. into integers, halves, quarters, etc. Is this why the distance between two numbers is defined as it is, since it only depends on the two numbers themselves and not on what is inbetween them?!

The reason I originally started pondering this was because I was considering an interval of numbers $(-10,10)$ which I partitioned up into intervals of $0.01$, i.e. such that $(-10,10)=(-10,-9.99,-9.98,\ldots ,0,0.01,0.02,\ldots ,9.98,9.99,10)$. Naively, I thought that there would be $2000$ numbers in this interval, but on further analysis I found that there were $2001$...

 

[Mark44]

So, is the intuitive answer simply that the distance between two points along the real number line should not depend on how we separate up the distance between them - it is an intrinsic property. It is not well-defined to define the distance between two real numbers in terms of counting the numbers between them, since the answer would depend on how we partition up the interval, e.g. into integers, halves, quarters, etc. Is this why the distance between two numbers is defined as it is, since it only depends on the two numbers themselves and not on what is inbetween them?!

Between any two distinct real numbers a and b, there are an infinite number of numbers, so it makes no sense to try to count the real numbers between a and b. This is a consequence of the reals being dense. The same is true for the rational number -- between any two distinct rationals there are an infinite number of rational numbers between them.

 

[Frank Castle]

Between any two distinct real numbers a and b, there are an infinite number of numbers, so it makes no sense to try to count the real numbers between a and b. This is a consequence of the reals being dense. The same is true for the rational number -- between any two distinct rationals there are an infinite number of rational numbers between them.

Is this the reason for defining the distance between two numbers as $\lvert a-b\rvert$?

 

[Mark44]

Is this the reason for defining the distance between two numbers as $\lvert a-b\rvert$?

Yes - the length of the interval between the two points.

 

[Frank Castle]

Yes - the length of the interval between the two points.

I guess what's messed me up is the intuitive picture that I picked up at school that the length of the interval between a number $a$ and $0$, i.e. $\lvert a-0\rvert=\lvert a\rvert$ is the number of units $a$ is from $0$, in this case $a$-units. But perhaps this is too much of a simplification?!

 

[TeethWhitener]

The same is true for the rational number -- between any two distinct rationals there are an infinite number of rational numbers between them.

Technical aside: is this true? The rationals have measure zero in the reals.

 

[Mark44]

I guess what's messed me up is the intuitive picture that I picked up at school that the length of the interval between a number $a$ and $0$, i.e. $\lvert a-0\rvert=\lvert a\rvert$ is the number of units $a$ is from $0$, in this case $a$-units. But perhaps this is too much of a simplification?!

No it's not. There's a difference between the length of an interval in some units and the number of points between two points, if I'm understanding you correctly.
The distance between 1 and 3 is 2 "1 units" or 4 "1/2 units" and so on, which is very different from what you said in post 7: "counting the numbers between them". Possibly what you meant differs from what you actually said.

 

[Mark44]

Technical aside: is this true? The rationals have measure zero in the reals.

Yes, but they are still dense in the number line. Between any two rational numbers there are an infinite number of other rational numbers. In contrast, the integers are not dense in the number line.

 

[TeethWhitener]

Between any two rational numbers there are an infinite number of other rational numbers.

Right. But the rationals are countably infinite. I don't know that you can use the same notion of distance in the rationals that you can use in the reals.
EDIT: To be clear, it's absolutely true that the rationals are dense in the reals, but specifically, I'm hesitant to talk about the concept of "distance" over a set of measure zero using the same metric that you would use over a set of non-zero measure.

 

[Frank Castle]

The distance between 1 and 3 is 2 "1 units" or 4 "1/2 units" and so on

So is the point that the distance between two numbers is intrinsic, but its numerical value depends on the unit of distance that we choose as a basis (e.g. "1 units" or "1/2 units")?

what you said in post 7: "counting the numbers between them". Possibly what you meant differs from what you actually said.

Apologies, I didn't word this very well in post #7. What I meant is that I wanted to determine the number of elements in the (ordered) set $\lbrace -10,-9.99,-9.98,\ldots ,0,0.01,0.02,\ldots ,9.98,9.99,10\rbrace$. Naively, I just worked out the distance between 10 and -10, which was of course 20, and then, as my unit of distance (between consecutive points) is 0.01, this gives 20x0.01=2000 points. But it turns out that I didn't take into account the end point (-10), which gave 2001 points in total.

 

[Mark44]

So is the point that the distance between two numbers is intrinsic, but its numerical value depends on the unit of distance that we choose as a basis (e.g. "1 units" or "1/2 units")?

I don't know if the right word is "intrinsic". There are different metrics that can be used to measure distance. For example, in R², there's the Euclidean norm, with $d(x, y) = \sqrt{(x_2 - x_1 )^2 + (y_2 - y_1 )^2}$ and there's also the so-called "taxicab" norm, with $d(x, y) = |x_2 - x_1| + |y_2 - y_1|$, just to name two of them. See https://en.wikipedia.org/wiki/Norm_(mathematics).

Apologies, I didn't word this very well in post #7. What I meant is that I wanted to determine the number of elements in the (ordered) set $\lbrace -10,-9.99,-9.98,\ldots ,0,0.01,0.02,\ldots ,9.98,9.99,10\rbrace$. Naively, I just worked out the distance between 10 and -10, which was of course 20, and then, as my unit of distance (between consecutive points) is 0.01, this gives 20x0.01=2000 points. But it turns out that I didn't take into account the end point (-10), which gave 2001 points in total.

This is the classic "fence-post" error.

If you have fenceposts every 10 feet, starting at 20 ft up to 110 ft, there are $\frac{110 - 20}{10} + 1 = 9 + 1 = 10$ fenceposts.

 

[Frank Castle]

I don't know if the right word is "intrinsic".

I guess what I meant by this is that, given a particular metric, the distance between two real numbers $a$ and $b$ is independent of the units that we use. For example, in the case I gave, given the metric $d(a,b)=\lvert a-b\rvert$, the distance between $a=10$ and $b=-10$ is always $d(10,-10)=\lvert 10-(-10)\rvert =20$ units, but the value can change depending on what one chooses one's units to be. In this example I further divided up each unit into intervals of 0.01, in which case the distance between -10 and 10 is 2000 "0.01 units".

Is it correct to intuitively think of a real number $x$ as being a distance of $\lvert x\rvert$ units from $0$?

There are different metrics that can be used to measure distance.

Is it the case then, that for $\mathbb{R}$ one usually chooses $d(x,y)=\lvert x-y\rvert$ as the metric (which I guess is simply the one dimensional case of the Euclidean metric)?!

This is the classic "fence-post" error.

If you have fenceposts every 10 feet, starting at 20 ft up to 110 ft, there are 110−2010+1=9+1=10\frac{110 - 20}{10} + 1 = 9 + 1 = 10 fenceposts.

What is the intuitive reason for why this error arises?

 

[Mark44]

I guess what I meant by this is that, given a particular metric, the distance between two real numbers $a$ and $b$ is independent of the units that we use. For example, in the case I gave, given the metric $d(a,b)=\lvert a-b\rvert$, the distance between $a=10$ and $b=-10$ is always $d(10,-10)=\lvert 10-(-10)\rvert =20$ units, but the value can change depending on what one chooses one's units to be. In this example I further divided up each unit into intervals of 0.01, in which case the distance between -10 and 10 is 2000 "0.01 units".

Is it correct to intuitively think of a real number $x$ as being a distance of $\lvert x\rvert$ units from $0$?

Yes

Is it the case then, that for $\mathbb{R}$ one usually chooses $d(x,y)=\lvert x-y\rvert$ as the metric (which I guess is simply the one dimensional case of the Euclidean metric)?!

Yes

What is the intuitive reason for why this error arises?

(Re: fencepost error) The intuitive reason is forgetting to count the starting fencepost. The distance from 0 to 10 is (obviously) 10 units, but if you have fenceposts at 0, 1, 2, 3, ..., 9, 10 feet, there are 10 + 1 = 11 fenceposts.

 

[Frank Castle]

(Re: fencepost error) The intuitive reason is forgetting to count the starting fencepost. The distance from 0 to 10 is (obviously) 10 units, but if you have fenceposts at 0, 1, 2, 3, ..., 9, 10 feet, there are 10 + 1 = 11 fenceposts.

Ah ok, so it is simply the case that one has forgotten to include start point (at which the first unit interval starts).

Is there any intuitive reasoning/ motivation for defining the distance between two real numbers as it is? Does it simply follow from geometrical analysis - thinking of the real numbers as lying along a geometric line (the real number line) and then using Pythagoras's theorem in one dimension?!

Also, does one take a unit interval (from 0 to 1), i.e. 1, as the basis for defining the distance between two real numbers? By this I mean, for example, the number 10 is a distance of 10 units from 0. It is always a distance of 10 units from 0 regardless of whether I split the interval between 0 and 10 up further into fractions of one unit.

 

[micromass]

Right. But the rationals are countably infinite. I don't know that you can use the same notion of distance in the rationals that you can use in the reals.
EDIT: To be clear, it's absolutely true that the rationals are dense in the reals, but specifically, I'm hesitant to talk about the concept of "distance" over a set of measure zero using the same metric that you would use over a set of non-zero measure.

Distances and measures are two separate worlds. The rationals are definitely a metric space but one with measure zero.

 

[TeethWhitener]

The rationals are definitely a metric space but one with measure zero.

Can't we say the same thing about the integers? Is the fact that the rationals are dense important to the notion of their being a metric space?

 

[micromass]

Can't we say the same thing about the integers? Is the fact that the rationals are dense important to the notion of their being a metric space?

Yes, the integers form a metric space too with the distance $|a-b|$.

 

[TeethWhitener]

Thanks for the clarification

 

[S.G. Janssens]

Distances and measures are two separate worlds. The rationals are definitely a metric space but one with measure zero.

To anyone interested in this topic (i.e. the relationship between measure-theoretic and (metric) topological concepts) I would like to recommend Oxtoby's "https://www.amazon.com/dp/0387905081/?tag=pfamazon01-20". It is both short as well as understandable. I find this a pleasant combination.

 

[Frank Castle]

Yes, the integers form a metric space too with the distance |a−b||a-b|.

When defining a distance between two real numbers are we essentially taking advantage of the one-to-one correspondence between real numbers and geometric points along the so-called real number line. By choosing an origin $\mathcal{O}$, which we identify with the real number $0$, then the interval between the origin and the point that we identify with the first integer $1$ defines a length scale, and the length of this interval is what we define as our length scale - it is a unit of length. Then, any other real number $x$ will be a distance of $\lvert x\rvert$ units from $0$. Given this, we note that one unit is the length of the interval between any two consecutive integers, and we then define the length between any two real numbers $x$ and $y$ as $$d(x,y)=\lvert x-y\rvert =\lvert y-x\rvert$$ which in words, intuitively means that "the distance between the real numbers $x$ and $y$ is $\lvert x-y\rvert$ units" (or in other words, "the real number $x$ is at a distance of $\lvert x-y\rvert$ units from the real number $y$").
Given this, if one further partitions each unit interval into sub-intervals of 0.01 units (i.e. 0.01xlength of a unit interval), then the numerical value of the distance between two real numbers $x$ and $y$ is re-scaled, such that the length between them is $\frac{\lvert x-y\rvert}{0.01}$ "0.01 units".

Would this be a correct understanding?

 

[jbriggs444]

When defining a distance between two real numbers are we essentially taking advantage of the one-to-one correspondence between real numbers and geometric points

The "distance" between two real numbers does not depend at all on the number of real numbers between them. Cast that thought out of your head. It is both irrelevant and incorrect.

That said, we do choose the distance between numbers on the real number line to be consistent with the distance between corresponding integers on the integer line. That is, we choose to define things so that the distance between 1.0 and 2.0 is the same as the distance between 1 and 2.

Given this, if one further partitions each unit interval into sub-intervals of 0.01 units (i.e. 0.01xlength of a unit interval), then the numerical value of the distance between two real numbers $x$ and $y$ is re-scaled, such that the length between them is $\frac{\lvert x-y\rvert}{0.01}$ "0.01 units".

The distance between two real numbers depends only on how you define distance, not on how you partition intervals.

I can measure the distance between New York and Boston in miles, in kilometers, inches or furlongs. Whether I drive straight through or stop three times out of deference to my wife's bladder capacity, it does not change the total distance traveled.

 

[Frank Castle]

The distance between two real numbers depends only on how you define distance, not on how you partition intervals.

I guess my confusion lies in the fact that one often speaks of a number $x$ being $\lvert x\rvert$ units from the origin, $0$ (for example, the number $10$ is a distance of $10$ units from $0$), or there being $\lvert x-y\rvert$ units between the two numbers $x$ and $y$ (for example, there is a distance of $0.97$ units between $3.22$ and $1.25$) . My question is, do we define the distance between two real numbers in terms of the length of the interval between $0$ and $1.0$, referring to this as a unit, and then, in this sense, the distance between two real numbers is independent of how we partition the unit intervals between them?!

 

[jbriggs444]

I guess my confusion lies in the fact that one often speaks of a number $x$ being $\lvert x\rvert$ units from the origin, or there being $\lvert x-y\rvert$ units between the two numbers $x$ and $y$. My question is, do we define the distance between two real numbers in terms of the length of the interval between $0$ and $1.0$, referring to this as a unit?!

Yes, we usually do. It is a simple metric with desirable properties like the fact that it is independent of the choice of origin.

But that does not mean that it arises out of some kind of "count the sub-intervals" rule.

 

[Frank Castle]

Yes, we usually do. It is a simple metric with desirable properties like the fact that it is independent of the choice of origin.

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 $y$".

 

[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.