# An Impossible Ratio

I'm facing a bit of a difficulty thinking about the aspect ratio of A4 paper.

The beauty of this paper size is that when it is folded in half along the longer side, it becomes A5 paper which has dimensions proportional to A4 paper. (ie, the aspect ratio of all A series paper is constant)

Let $a$ be the longer side of my A4 paper and $b$ be the other. If I fold it in half along the width $a$, then that side becomes $\frac{1}{2}a$ which is now less than the other side $b$.

Now, $b$ is the long side and $\frac{1}{2}a$ is the short side.

So, since the length to width ratio of these rectangles are the same because of what I've mentioned as the beauty of A4 paper : $$\frac{a}{b} = \frac{b}{\frac{1}{2} a} \implies \left|\frac{a}{b}\right| = \sqrt{2}$$

How is it possible an incommensurable quantity be expressible in the real world units?

Edit: I was under the assumption that $a,b \in \Bbb Z$ (set of integers) because in the real world we are not so accurate as to have irrational lengths. But upon further reflection, we can't have any non-terminating decimal exist in real world units either because of limitations of precision.

(Question inspired by Numberphiles)

• Lesson-learnt: A ratio need not be commensurable but real-world aspect ratio is commensurable due to a lack in precision of cutting instruments. – Nick Nov 8 '14 at 13:06
• The real world is not so accurate as to have exactly rational lengths either ... – hmakholm left over Monica Nov 8 '14 at 13:07
• @HenningMakholm: Ah yes, repeating non-terminating decimals such as $\frac{1}{3}. \frac{1}{7}, \frac{1}{9}$ etc. – Nick Nov 8 '14 at 13:20
• x @Nick: There's nothing about $\frac{1}{10}$ that makes it easier for it to exist in the real world than $\frac{1}{9}$. – hmakholm left over Monica Nov 8 '14 at 13:21
• @HenningMakholm: I hope you're not talking about mathematical nominalism. What you say does not make sense without units. In human friendly scale of length, $0.1 \text{ cm} = 1 \text{ mm}$. We see it existing. Don't we? – Nick Nov 8 '14 at 13:53

Why wouldn't it be possible?

It is well known that in a plane that satisfies the usual axioms of plane geometry, there are lines that have exactly this ratio -- for example the ratio of the diagonal of a square to its side. And it is easily possible to put two such lines at right angles to each other.

Now in actual industrial fact, the dimensions of the A-series paper formats are defined by numbers rounded to whole millimeters (with tolerances measured in millimeters too), so the "all formats have the same ratio" slogan is only approximately true. But that's more or less independent of the underlying mathematical model which has no problems with incommensurable ratios whatsoever.

• An incommensurable number means a number that cannot be expressed as a ratio of integers. Okay, I get it, a ratio need not be expressed interms of integers. But are you saying that the fundamental ratio involved here is $a:b \equiv \sqrt{2}:1$ – Nick Nov 8 '14 at 12:51
• @Nick: Yes, $\sqrt2:1$ is a perfectly cromulent ratio. – hmakholm left over Monica Nov 8 '14 at 12:58
• You say that the constant aspect ratio of A-series paper is an approximation. What is the approximate error I should expect during resize of content? – Nick Nov 8 '14 at 13:10
• @Nick: Around the same (or less) as for the $\pm 1$ mm tolerance in the physical dimensions of the sheets. – hmakholm left over Monica Nov 8 '14 at 13:16

Why should a ratio not be incommensurable? For example, the ration of the diagonal to the side of a square is also $\sqrt 2$. Incidentally, according to DIN 476 / DIN EN ISO 216 the side length of A4 paper are commensurable because thay are defined to be $210\times 297$ millimeters (so $\sqrt 2$ is only an approximation to the norm, or is the motivation behind the norm). Then again, even the slightest tolerance allowed makes any distinction rational vs. irrational fruitless.

• Oh, incommensurable ratios are only a subset of the set of all ratios. Ok, so the ratio here is $\sqrt{2}:1$ – Nick Nov 8 '14 at 12:53

You might be interested in reading the wikipedia article about constructible numbers: https://en.wikipedia.org/wiki/Constructible_number

That is, numbers which can be constructed on a piece of paper with unruled straightedge and compass....

• Indeed I am. Thank you very much but this does not answer my question. Please leave your answer as a comment. This post is now unnecessary. – Nick Nov 16 '14 at 23:05

In addition to the specification that the ratio of sides is $\sqrt2$, the area of the $\text{A}0$ paper is $1m^2$ thus determining the dimensions of the A series of paper.

One wonders if there is might be an analogy in $\Bbb R^3$ for this ratio.

• an analogy in $\Bbb R^3$. explain. – Nick Nov 16 '14 at 23:06
• Halving a cuboid, for instance. – hypergeometric Nov 17 '14 at 14:43

Geometry -- the mathematical subject -- describes not our actual world of experimentally measured space, time and matter, but instead an idealized, make-believe world which runs on very different physics. In that far-away place, any quantity can be measured with unlimited precision, and writing down the infinitely extended results of such measurements takes no time at all. The lucky inhabitants are able to transmit information with infinite bandwidth, so they can send each other the necessary non-repeating decimal numbers instantly.

In this Shangri-La of Mathematical life, any person can effortlessly and instantly tell the difference between the number $2^{1/2}$ and $1.41421356237$... (insert an unlimited number of digits here). As a result, the square root of $2$ and any finite approximation to it are completely and unambiguously different. They are, in a word, incommensurable!

But in our less-than-perfect universe, every distance we measure and use in everyday life is rationally approximated in a very few digits. Non-repeating decimals never appear.

If you live in the Paradise of Math, incommensurability is likely to cause you some significant difficulties. But in our world? There is nothing to worry about on that score. Incommensurability happens only in the mind.