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A friend sent me a link to this item today, which is billed as an "Irrational Numbers Wall Clock."

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There is at least one possible mistake in it, as it is not known whether $\gamma$ is irrational.

Anyway, this started me wondering about how to improve the design of this clock from a mathematical standpoint. Here's my formulation of the problem:

  1. Find 12 numbers that have been proved to be irrational to place around the outside of a clock.

  2. Each of eleven of the numbers must approximate as closely as possible one of the integers from 1 through 11. The 12th can either be just smaller than 12 or just larger than 0.

  3. The numbers must have expressions that are as simple as possible (in the spirit of - or even more simple than - those in the clock given in the picture here). Thus, for example, no infinite sums, no infinite products, and no continued fractions. Famous constants and transcendental functions evaluated at small integers encouraged.

  4. Expressions should be as varied as possible. Better answers would include at least one use of trig functions, logarithms, roots, and famous constants.

Obviously, goals 2, 3, and 4 act against each other. And, as Jonas Meyer points out, "as closely as possible" and "as simple as possible" are not well-defined. That is intentional. I am afraid that if I tried to define those precisely I would preclude some answers that I might otherwise consider good. Thus, in addition to the mathematics, there's a sizable artistic component that goes into what would be considered a good answer. Hence the "soft-question" tag. I'm really curious as to what the math.SE community comes up with and then what it judges (via upvoting) to be the best answers, subject to these not-entirely-well-defined constraints.

Note that the designer of the clock given here was not trying to approximate the integers on a clock as closely as possible.

Finally, it's currently New Year's Day in my time zone. Perhaps a time-related question is appropriate. :)

Note: There is now a community wiki answer that people can edit if they just want to add a few suggestions rather than all twelve.

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"As closely as possible" and "as simple as possible" are not defined. What's to stop you from taking rational powers of $2$ for each? –  Jonas Meyer Jan 1 '11 at 23:20
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As Jonas points out, variety might be nice. I'd suggest a #4: Expressions should ideally be as varied as possible (e.g., at most one root, at most one use of a trig function, etc.). –  Matthew Conroy Jan 1 '11 at 23:29
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Here's another (ahem) "timely" use of irrationals: piclock.com :) (Note: That's my product. I'll delete this comment if anyone objects to my mentioning it here.) –  Blue Jan 2 '11 at 0:22
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For $7$, how about $22/\pi$? By the way, I think this would work well as a community wiki question. –  Jonas Meyer Jan 2 '11 at 1:56
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I might add a bias towards over-estimates. That way, should anyone care to memorize the expressions, they can impress their friends by saying things like "e-to-the-phi is 5-point-something, I just forgot the 'something' at the moment". :) –  Blue Jan 2 '11 at 5:03

8 Answers 8

up vote 21 down vote accepted

Here's a list of irrational numbers which almost fulfill your criteria.

Each were chosen to be accurate to within ±0.1, so that no two of them implicitly express the same mathematical approximation, and so that none of them "cheats" in order to fudge an exact result involving integers to obtain a slightly inexact, irrational result. Only the last number fails to meet your criteria, as it is slightly larger than 12.

  1. $\ln(3)$
  2. $7\pi/11$
  3. $\sqrt 2 + \frac\pi2$
  4. $7/{\sqrt[3]5}$
  5. $\mathrm e^\phi$
  6. $\sec^2(20)$
  7. $4\sqrt3$
  8. $5\phi$
  9. $2\pi+\mathrm e$
  10. $\sinh(3)$
  11. $\pi^3-20$
  12. $\csc^2(16)$

I would prefer not to use cosecant, integers greater than 12, or more than two additions/subtractions — it is too easy to get results if you rely on these — but I think I've spent enough time on this diversion for now. :-)

[EDIT: revised the formula for 4 two times now: this first to change the formula for 4 from √3 + √5 — which is too close to √4 + √4 — and the second time to correct the formula as I somehow copied a result which was not approximately 4.]

Added: Since it seems that I can't sleep tonight, here is a list of approximate values:

$\begin{align*} \ln(3) & \approx 1.0986 \\ 7\pi/11 & \approx 1.9992 \\ \sqrt 2 + \tfrac\pi2 & \approx 2.9850 \\ 7 / \sqrt[3]5 & \approx 4.0936 \\ \mathrm e^\phi & \approx 5.0432 \\ \sec^2(20) & \approx 6.0049 \\ 4\sqrt3 & \approx 6.9282 \\ 5\phi & \approx 8.0902 \\ 2\pi+\mathrm e & \approx 9.0015 \\ \sinh(3) & \approx 10.018 \\ \pi^3-20 & \approx 11.006 \\ \csc^2(16) & \approx 12.064 \end{align*}$

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Nice! Could you add the approximate values? –  Mike Spivey Jan 2 '11 at 3:48
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Sure! The value of each expression is now displayed with its decimal value (rounded to the nearest multiple of 1/5). ;-) –  Niel de Beaudrap Jan 2 '11 at 4:02
    
Good one, Niel. :) :) I meant so that I could see just how close each number is to the integer it's supposed to be approximating (more precisely than the 0.1 you've already mentioned). –  Mike Spivey Jan 2 '11 at 4:09
    
I've made a correction, and added the approximate values. –  Niel de Beaudrap Jan 2 '11 at 4:50
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Is $2\pi+e$ known to be irrational? –  simplequestions Jan 2 '11 at 13:34

Here are my meager suggestions:

  • For a new "zero", the symobl $\epsilon$, which although not representing a specific number, is nearly universally used to represent a very small positive quantity.

  • Alternatively, for zero one could use $\frac{1}{\omega}$, which is one of the most canonical infinitesimals in the surreal numbers.

  • For 2, one could use $|1+i|^2$. (I realize this is exact and hence rational, but the expression involves non-rational numbers.)

As a mathematician who is also an antique clock collector, I love this question, and I would be interested if after some good answers are submitted, they are collected and made into a suitably attractive pdf image that could be printed and actually used for a clock face. My suggestion would be to adopt an old-style clock typography, if this is possible, since it might also help the clock face look more like a clock face.

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Thanks. I particularly like the use of $\epsilon$ for an approximation of 0. –  Mike Spivey Jan 2 '11 at 3:54
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Your "2" wastes a square that might be used elsewhere. Instead, you might opt for $5 = |3+4i|$. This, too, has the rationality problem; on the other hand, tweaking to something like $8 \approx |5+6i|$ seems like a "cheat" in @Niel's sense. (BTW, I also like $\epsilon$ for $0$.) –  Blue Jan 2 '11 at 4:52
    
Day Late, I agree. My preference would be to find completely natural instances, using only very low integers and perhaps avoiding + completely (except perhaps for complex numbers?). I don't see it as a problem if the expressions are rational or even integer-valued, provided that they are mathematically interesting. –  JDH Jan 2 '11 at 5:32

This is intended to be a community wiki answer that people can edit if they just want to give a few numbers rather than all twelve. I'll start with suggestions currently in the comments. Feel free to add more!

For 1: $ -\sin 11$

For 3: $e + \sqrt{\frac{5}{63}}$

For 7: $22/\pi$

For 12: $\log_{1922}(1782^{12}+1841^{12})$ (Approximately 11.99999999996, featured on The Simpsons.)

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I wonder if it is cheating to use both $-\sin(11)$ and $22/\pi$, considering that $\sin(11)$ is close to $-1$ because $11$ is close to $\frac{7\pi}{2}$. –  Jonas Meyer Jan 2 '11 at 4:31
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I really like the one for 12. –  Raskolnikov Jan 2 '11 at 12:27
    
@Jonas: Feel free to remove one of them. After all, they were both suggestions of yours! –  Mike Spivey Jan 3 '11 at 5:50
    
Good point :) I don't want to be too hard on myself. I was just wondering out loud after reading Niel's point about answers that implicitly express the same mathematical approximation. It's probably in the spirit of this thread to include redundant answers for the time being, and to take the best ones in the end. –  Jonas Meyer Jan 3 '11 at 7:09

Suggestions

11: $e^{\pi i}$

4: ${\sqrt 5}^{\sqrt 3}$ (aprox. 4.03019240+ )

While, it's no match for the previous entry, I can't resist giving one more for

11: $\frac{\sqrt[4]{104 + e^{\pi \sqrt{58}}}}{6^2}$

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$11 = -1 \mod 12$. Nice. :) –  Mike Spivey Feb 26 '11 at 0:26

Despite the large constant, I like tanh(2011) for 1. It is very close and shows the year.

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That could also be an idea, only using the number 2011 and elementary operations and transcendenatal functions try to construct 12 approximations to the the numbers from 1 to 12. –  Raskolnikov Jan 2 '11 at 12:37

Some that I like:

  • I'm rather fond of $3 \approx \log 20$. Every so often I find myself taking large powers of $e$ in my head; knowing this is helpful.

  • $7 \approx 12 \log_2 (3/2)$. This expresses the fact that, in music, a fifth is seven-twelfths of an octave.

  • $6 \approx \log (\pi^4 + \pi^5)$; I've actually seen T-shirts claiming that $\pi^4 + \pi^5 = e^6$.

  • $\pi + \pi^2 \approx 13$, although unless we're in an Orwell book your clocks have no thirteen.

The wikipedia article on mathematical coincidences may have more ideas.

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+1 for the 6 –  ypercube Feb 25 '11 at 23:46
    
You could write the 3 as $\log 10 + \log 1010$ –  ypercube Feb 25 '11 at 23:49
    
Nice answers (especially the 6). Thought you might want to know that AcidFlask mentioned the "mathematical coincidences" site as well. –  Mike Spivey Feb 26 '11 at 0:25
    
@Mike: I just did. Had to wait 40 minutes as I overused my votes yesterday :) –  ypercube Feb 26 '11 at 0:34
    
@ypercube: Fair enough. My apologies for bugging you about it! I will delete my comment. :) –  Mike Spivey Feb 26 '11 at 0:38

After seeing the ThinkGeek :: Pop Quiz Math Clock a few years ago I asked my friend David a question related to the one here: to use more interesting mathematical approaches to generate each of the integers 1-12. He provided a great answer at http://www.astronexus.com/comment/reply/153

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Have a look at Wikipedia for some interesting ideas:

I like $\pi^2 \approx 9.87$. Fooling around with variations of what's there also yielded

$$ 7( \pi - e) \approx 2.96 $$

$$ 19 (\pi - e) \approx 8.05 $$

Furthermore the numbers don't have to be limited to (0, 12], since you could always work in $\mod 12$. For example $ (\pi + 20)^i \approx -1 $ could be used for 11 o'clock.

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