Meaning of division by non-whole real numbers

[logicgate]

TL;DR

How does the interpretation of division (a/b) as a divided equally into b equal parts each of size a/b work when b is a non whole real number like e/pi for example ?

We're all familiar with positive integer division like for example 12/3 mean 12 is split into 3 equal groups each of size 4. But what about division by non whole real numbers like pi divided by e ? Can we interpret it the same way as we did with 12/3 ? Can we interpret pi/e as pi is divided into e equal groups each of size pi/e ? Division by non whole reals doesn't make sense to me.

 

[A.T.]

Can we interpret pi/e as pi is divided into e equal groups each of size pi/e ?

You have e of pi/e in 1 pi, if that's what you mean.

Division by non whole reals doesn't make sense to me.

How would you compute the diameter of a circle from a known circumference?

 

[jbriggs444]

Are you comfortable with long division using pencil and paper? My generation was taught how to do this.

There is no problem in principle with carrying out this process using infinitely long decimal expansions. Just do it to ten decimal digits and compute ten (or so) digits of the quotient. Go back and extend all of those calculations to twenty decimal digits. And continue the long division until you have twenty (or so) digits of the quotient.

Repeat ad infinitum. That is one process.

Or, pretty much equivalently, truncate the dividend and divisor to ten digits each. Do an division and write down ten (or so) digits of quotient. Repeat, truncating to twenty digits each this time. Then thirty. And so on. The limit you approach is the true quotient.

If you are a mathematician there is a different approach that can be taken. One way of formally constructing the real numbers is as a set of equivalence classes of Cauchy sequences of rational numbers. [A "Cauchy" sequence is one in which all of the terms tend to end up all being arbitrarily close to one another -- for any positive epsilon (closeness) there is a delta (point in the sequence) beyond which all of the terms are within epsilon of one another].

Two Cauchy sequences are judged to be "equivalent" if one can interleave their terms and wind up with a Cauchy sequence.

Using this construction, pi is the equivalence class of Cauchy sequences that includes the exemplar: ( 3.0, 3.1, 3.14, 3.141, 3.1415, 3.14159, ...). Meanwhile, e is the equivalence class that includes the exemplar: (2.0, 2.7, 2.71, 2.718, 2.7182, 2.71828, ...).

One can define division for real numbers constructed in this manner by taking the limit of the term by term quotients of any two respective exemplars. For instance, pi/e would be the limit of (3.0/2.0, 3.1/2.7, 3.14/2.71, 3.141/2.718, 3.1415/2.7182, 3.14159/2.71828, ...)

A bit of care would need to go into worrying about division by zero and proving that every pair of exemplars yields an equivalent result.

Alternately and perhaps more conveniently one could work to first define multiplication and then define division as the inverse operation. I honestly cannot remember how we did it when I took that class. I think it was this way.

An alternate construction uses Dedekind cuts. The definition for multiplication is not difficult, but dealing with sign problems makes it somewhat inelegant for my taste. For the product of two positive real numbers, you basically form the lower cut from the set of products of non-negative rational pairs drawn from the lower cuts of the two factors. The upper cut is whatever positive numbers are left over. Division is then defined as the inverse of multiplication in the appropriate sense.

 

[FactChecker]

You can approximate it with a sequence of integer divisions. Long division by hand does that -- never going over and keeping track of the remainder, then continuing with an integer division of the remainder. You can get as close to the answer as you want.

 

[symbolipoint]

A simple and clear understanding occurred to me many many many years ago. Dividend, Divisor, Quotient. How many of the divisor are contained in the dividend, INCLUDING any fractional part?

I could say more, but at present I avoid doing so.

 

[PeroK]

TL;DR: How does the interpretation of division (a/b) as a divided equally into b equal parts each of size a/b work when b is a non whole real number like e/pi for example ?

We're all familiar with positive integer division like for example 12/3 mean 12 is split into 3 equal groups each of size 4. But what about division by non whole real numbers like pi divided by e ? Can we interpret it the same way as we did with 12/3 ? Can we interpret pi/e as pi is divided into e equal groups each of size pi/e ? Division by non whole reals doesn't make sense to me.

In pure mathematics it takes a lot of work to develop the real numbers. Integers and rational numbers are far simpler.

One of the properties of the real numbers is that every number (except zero) has a multiplicative inverse. That means that for every real number $x \ne 0$, there is another real number, $x^{-1}$ such that $x(x^{-1}) = 1$.

Division is no longer seen as a separate binary operation. Instead $x/y \equiv xy^{-1}$.

There is a case that Division of real numbers only really makes sense in this more abstract context.

 

[Roberto Pavani]

As a humble engineer, I think there's both a counting bias and a decimal bias here. We are taught to think of numbers mainly as counts of objects, and later as decimal expansions, so division tends to be associated with splitting things into a whole number of groups. But division by a real number often makes more sense as a change of units. For example, if the unit is a full circle, then dividing by $\pi$ gives 2, since one circle equals $2\pi$ radians. It's a measurement problem rather than a grouping problem.

 

[jbriggs444]

As a humble engineer, I think there's both a counting bias and a decimal bias here. We are taught to think of numbers mainly as counts of objects, and later as decimal expansions, so division tends to be associated with splitting things into a whole number of groups. But division by a real number often makes more sense as a change of units. For example, if the unit is a full circle, then dividing by $\pi$ gives 2, since one circle equals $2\pi$ radians. It's a measurement problem rather than a grouping problem.

Indeed. We teach elementary school students using integer ratios and decimal fractions. As a pure mathematician, I tend to shy away from decimals. But when communicating with laymen, it is handy to fall back on a shared background of decimal fractions that have only finite precision.

I am familiar with constructions of the real numbers using either Dedekind cuts and Cauchy sequences. Neither of those involve decimal expansions. Both do use integer fractions (the rational numbers) as a starting point. I am not familiar with any constructions that avoid that.

With my mathematical hat firmly in place, I do not think about measurement or the real world at all. Almost all "real numbers" are physically unrealizable.

 

[mathwonk]

@Roberto Pavani:
As a geometer, I share your point of view, as did Euclid. I.e. real numbers are essentially lengths on a line, (after choosing a unit length). One can also treat them as areas of rectilineal figures. In Prop. I.45, he shows how to construct a parallelogram (actually with given side) with area equal to that of another rectilineal figure. This allows one to define both products and quotients of lengths.
I.e. a rectangle can be thought of as a having area the product of its side lengths, and by finding another equal rectangle having one side of unit length, one can transform an area into a length. Similarly, given a rectangle of sides 1 and x, and another length y, constructing a rectangle with side y and equal to the original rectangle (in area) gives the quotient of x by y as the other side length.
Indeed our word "divides" is actually rendered in Euclid, even in the case of integers, by the word "measures", as used e.g. in the proof of Prop. IX.20.

This representation of real numbers as ratios of lengths of line segments was prevalent for centuries before the advent of algebraic and analytic versions such as that of Dedekind. Even Dedekind's approach is a sort of converse to that of Euclid, as made precise by Hilbert. I.e. Hilbert clarified Euclid's tacit assumption that every point on a line divides that line into two disjoint sides; Dedekind postulated the opposite, that any separation of a line into two disjoint (open) sides could only be achieved by removal of a point. I.e. points, or real numbers, can be defined as separations of the line.
An example of use by earlier scientists of pairs of line segments to denote real numbers occurs e.g. in Galileo's famous work On two new Sciences, Book III, where he uses them in his study of the motion of falling bodies.
Indeed Newton seems to have done the same in his Principia. Sadly my copy is long gone, but here is a link to such a usage:
https://en.wikisource.org/wiki/The_Mathematical_Principles_of_Natural_Philosophy_(1846)/BookI-I

A modern explanation of how this approach leads to decimal expansions of real numbers is given at the very beginning of vol. I of Richard Courant's Differential and Integral Calculus, pages 8-9.

 

[Matterwave]

@Roberto Pavani:
As a geometer, I share your point of view, as did Euclid. I.e. real numbers are essentially lengths on a line, (after choosing a unit length). One can also treat them as areas of rectilineal figures. In Prop. I.45, he shows how to construct a parallelogram (actually with given side) with area equal to that of another rectilineal figure. This allows one to define both products and quotients of lengths.

Genuine question here as I am not so familiar with this view of things. Would this bring into question constructable vs non-constructable numbers? I suppose if you simply posit that any length line is valid then this would define operations over the reals. But almost all real numbers are not constructable so if you actually want to be able to "draw" such a line or figure...

 

[mathwonk]

Interesting question. I haven't thought much about it, but if I understand, I think the point here is that, if one is given a unit segment, and also segments representing x and y, then from them one can construct segments representing both xy and x/y.

I.e. although given only a unit segment, the fact that one cannot construct from it segments representing either e or π, does not change the fact that if one is given such segments, then one can also construct from them segments representing both eπ and e/π.

Notice also that although one can quickly construct from a given unit segment, a segment representing sqrt(2), one cannot give any finite process producing its (full) decimal expansion.

 

[mathwonk]

Interesting question. I haven't thought much about it, but if I understand, I think the point here is that, if one is given a unit segment, and also segments representing x and y, then from them one can construct segments representing both xy and x/y.

I.e. although given only a unit segment, the fact that one cannot construct from it segments representing either e or π, does not change the fact that if one is given such segments, then one can also construct from them segments representing both eπ and e/π.

Notice also that although one can quickly construct from a given unit segment, a segment representing sqrt(2), one cannot give any finite process producing its (full) decimal expansion.

It is even easier to construct products and quotients of segments using similar triangles, i.e. triangles constructed within the same vertex angle, having parallel bases.

 

[Matterwave]

Yeah, I agree that if given segments x and y and 1, then we should be able to construct xy, x/y.

But I wonder if the ancients were so interested in figuring out constructable numbers for reasons similar to "I want to represent numbers with lengths or areas and I want to be able to draw these lengths and areas". This is more of a history of math question than a math question though.

 

[Roberto Pavani]

It depend how "ancients" note that negative numbers didn't exists because negative area was not understood. Even zero wasn't accepted in very ancient math

 

[PeroK]

Genuine question here as I am not so familiar with this view of things. Would this bring into question constructable vs non-constructable numbers? I suppose if you simply posit that any length line is valid then this would define operations over the reals. But almost all real numbers are not constructable so if you actually want to be able to "draw" such a line or figure...

One can postulate that nature can do real numbers and base their existence and divisibility from that.

However, most real numbers are literally indescribable. There would be no way to specify which number we are talking about - e.g. the length of an object.

Alternatively, one could argue that below a certain scale, the length of an object becomes ambiguous. Even without QM, where precisely does a metre stick end?

The only certainty is that the real numbers can be constructed mathematically and division can be defined purely mathematically.