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High school maths and ratios at KA show how to solve ratio problems, and when ratios are equivalent - but don't say what the ratio is.

I've worked out some canonical forms, and I wondered why something like this isn't used, when they seem to underlie what is taught. Have I got them correct, and is there a problem with the idea of using canonical forms for ratios?

  • Naturals: the canonical form of a ratio is $p:q$ with $p, q \in \mathbf{N}$ are in lowest terms. This canonical form isn't a number itself, but can only be expressed as a ratio of numbers.

    Another ratio is equivalent to a canonical if both sides are the same multiple of the canonical's sides i.e. $m.p:m.q$ with $m \in \mathbf{N}$. So $4:3$ is the ratio, and $8:6$ and $16:12$ are equivalent. But there's no $m$ to give $10$ on the left (and the "corect" right side, $7\frac{1}{2}$, wouldn't be natural anyway).

  • Rationals: a canonical form of the ratio $p:q$ is the rational number $\frac{p}{q}$, again with with $p, q \in \mathbf{N}$ in lowest terms. This canonical form is simply a rational number.

    These are the "same" ratios as for naturals, but with additional equivalent ratios, because we can multiply by a rational instead of by a natural. i.e. $\frac{m}{n}.p:\frac{m}{n}.q$ where $m,n \in \mathbf{N}$ - which is equivalent to multiplying, dividing, or multiplying and dividing by naturals.

  • Reals: the canonical form of a ratio $p:q$, where $p,q \in \mathbf{R}$ is simply the real quotient $\frac{p}{q}$. This canonical form is simply a real number.

    A ratio is equivalent to a canonical form if its quotient equals the canonical form.

(I've omitted negatives because they don't seem to be used. They are straightforward to add, but make it less clear.)

Wikipedia's article on ratios mentions using the real quotient - but that's not how it's taught.

Perhaps it's a hangover from Euclid, who presented integral ratios (Book VII, D20) separately from rational (and kinda irrational) ratios, (Book V, D5)? Though he doesn't use canonical forms either. (BTW the wiki ratio article has a nice overview, in History and etymology, Euclid's Definitions).

hyperpallium
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    canonical forms for ratios Define "canonical form". Certain forms make better sense in certain contexts, for example there is no clear universal "winner" between $1 / \sqrt{2}$ vs. $,\sqrt{2} / 2,$. Reals: ...simply the real quotient But reals include all rationals, which in turn include all integers, so this contradicts the previous points. Also, $,4 \pi / 2 \pi = 2,$. – dxiv Feb 16 '18 at 04:46
  • @dxiv a canonical form is a form to which all instances can be converted uniquely (uniquely in that, for each instance, there is only one possible canonical form). Alt: a nominated instance of a set of equivalent instances. OK, thanks, I see canonical forms might cause disagreement in gemeral, but here specifically, "lowest terms" and "quotiemt" are clear winners, aren't they? e.g. isn't a rational in lowest terms already accepted as a canonical form? – hyperpallium Feb 17 '18 at 02:02
  • isn't a rational in lowest terms already accepted as a canonical form Generally so, with the denominator usually assumed to be positive. But you run into far worse ambiguities once you step into the reals and beyond. – dxiv Feb 17 '18 at 02:07
  • @dxiv Yes, you can use the real canonical form for naturals (etc), but that doesn't prevent another canonical form defined just for integers, does it?. I was treating them separately as that is how they seem to be taught. A separate definition for each domain, like integer addition being closed over the integers or functions of different domains, That is, there are integer_ratios, rational_ratios, and real_ratios. – hyperpallium Feb 17 '18 at 02:17
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    Real ratio obviously exist. What I am not clear on is what you'd propose as a "canonical form" for them. – dxiv Feb 17 '18 at 02:23
  • @dxiv So, according to these definitions, $4\pi:2\pi$ isn't an integer_ratio. To support this notion, Euclid also has an integer-only definition of ratios (7,D20, linked in Q) (Though I agree that having several different "canonical forms" isn't very "canonical".) But I think I can read your reply as saying, in general, canonical forms have problems, so there's a general reluctance to use them by mathematicians - even if they'd be OK for this specific case of ratios. And even though I think I can answer your specific objections, they add complexity, and aren't as clear as I'd thought. – hyperpallium Feb 17 '18 at 02:24
  • [sry, we crossed replies - all of mine have been to your first comment] – hyperpallium Feb 17 '18 at 02:24
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    I think I can read your reply as saying, in general, canonical forms have problems, so there's a general reluctance to use them by mathematicians No, that's not what I meant. Canonical forms do in fact exist in many legitimate cases. I just don't see how a canonical form for arbitrary real ratios would be defined. – dxiv Feb 17 '18 at 02:32
  • @dxiv what you'd propose as a "canonical form" for [real ratios] Oh, I see. A "canonical form" is usually a syntactical construct, i.e. how it is written. But I was thinking of a ratio of two reals e.g. $\pi : e$ being represented by the single real of their quotient e.g. $\frac{\pi}{e}$ (the real number it evaluates to, that we can't notate directly). Now I see why you asked me to define "canonical form"! – hyperpallium Feb 17 '18 at 02:33
  • I had checked this part of the definition with wikipedia for a class of objects on which an equivalence relation is defined, a canonical form* consists in the choice of a specific object in each class.*, but unfortunately skimmed all its other parts and examples of "presentation". I can still read it with my meaning, but that's not the usual meaning. Is there a term for what I mean? Maybe, "canonical instance"? – hyperpallium Feb 17 '18 at 02:43

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