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

[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.

 

[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?