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.