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Walking with my son at 3:14pm the other day, I mentioned to him, "Hey, it's Pi Time". My son knows 35 digits of $\pi$ (don't ask), and knows that it's transcendental. He replied, "is it exactly $\pi$ time?"

This led to a discussion about whether there is ever a time each afternoon that is exactly $\pi$, meaning 3:14:15.926535...

This feels like some kind of Zeno's Paradox. I told him that (assuming time is continuous) it had to be $\pi$ time at some point between 3:14:00 and 3:15:00, but the length of that moment was 0. However, this discussion left him confused.

Can anyone suggest a good way to explain this to a child?

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Fixee
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    It's not really "Zeno's paradox"... More like taking photographs of someone walking along a river. If he is on one side of the river in the first series of photographs, and later in the series he is on the other side of the river, we deduce that somewhere along the line he must have crossed the river. 3:14:00 is a time on "one side of" $\pi$, and 3:15:00 is a time on "the other side of" $\pi$. Or try it with a graph: if the value is negative at $3$ and positive at $4$, and you have to draw the graph without lifting the pencil, somewhere along the line you went through the axis. – Arturo Magidin Sep 09 '11 at 21:11
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    Of course, if time is quantized, then the answer is that it is never exactly $\pi$ time, since $\pi$ is incommeasurable with $1$ second, and whatever length a quantum of time might be, it is commeasurable with $1$ second... – Arturo Magidin Sep 09 '11 at 21:12
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    Every number has an infinite decimal representation (just append 0's forever). So, is it ever exactly "1 time" or "2 time"? (I think "Yes", as per Arturo's first argument, but I just wanted to point out that $\pi$ is not special in this regard.) – Austin Mohr Sep 09 '11 at 21:18
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    @Arturo: Why should a quantum of time be commensurable with 1 second? – TonyK Sep 09 '11 at 21:28
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    @TonyK: Presumably, because 1 second is observable, and so must be an integral multiple of whatever the quantum of time is... – Arturo Magidin Sep 09 '11 at 21:30
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    @Arturo: I just looked it up: a second is 9,192,631,770 times the period of the radiation emitted by the transition between the two hyperfine levels of the ground state of the caesium 133 atom. So it looks like you're right! – TonyK Sep 09 '11 at 21:40
  • Somewhat related (though a different unit of measure) http://physics.stackexchange.com/questions/13426/is-anything-actually-1-meter-long-or-1kg-of-weight – The Chaz 2.0 Sep 09 '11 at 21:44
  • @TonyK, but in quantum mechanics, a wave of disturbances in an infinite square lattice can propagate with a wavelength that is not commensurable with the lattice spacing. So, by analogy, quantized time would not necessarily demand that the Cs-133 hyperfine transition frequency is commensurable with the time quantum. – hmakholm left over Monica Sep 09 '11 at 21:47
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    It seems there is a Pi time, but no Pi day. – Srivatsan Sep 09 '11 at 21:52
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    And when is hammertime? – The Chaz 2.0 Sep 09 '11 at 21:55
  • @Henning: Infinite square lattices are very expensive to build! So maybe we will never get to observe one. – TonyK Sep 09 '11 at 22:01
  • @TonyK: if time is quantized and a second is 9,192,631,770 times the period of that particular Cesium 133 emission, is the period of that particular Cesium 133 emission an integral multiple of the time quantum, or might that period itself be a mean of a range of time quanta? – robjohn Sep 09 '11 at 23:04
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    @The Chaz: Every second of every day. –  Sep 09 '11 at 23:04
  • You may show him a clockwatch (or any clock where the hands move continuously) to show the difference between a moment in time and a duration. A moment is just a point somewhere on the dial (the only difference between $1$ and $\pi$ is that there is no mark on the dial for the latter), the hand does not stay there, whereas a duration is the length between two points. But most clocks move once every second (because we usually don't need more precision) so we are led to confuse a mark on the dial with the second between two marks. – Joel Cohen Sep 10 '11 at 01:05
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    I think $\pi$ time should be 3:08:29.733552923255... – lhf Sep 10 '11 at 01:28
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    @Arturo: I tried the river example with my son; he responded, "he might have traveled around the Earth to the other side of the river." – Fixee Sep 10 '11 at 04:23
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    @Fixee: Well, in that case, I think the first thing you need to do is explain the nature of "metaphor" and "analogy" to him, then come back to math. – Arturo Magidin Sep 10 '11 at 04:31

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Pi time is not like a concrete slab on the sidewalk that you can stand on; it's not even like the crack between the slabs, which have a width. It is like the line precisely down the middle of that crack. When you're walking on the sidewalk, you cross right over it without stopping on it.

And so is every other precise time: like exactly noon or midnight.

Of course, the analogy fails a little bit because when you stop on the sidewalk, you cover a whole range of positions. As far as we can tell in everyday life, that's not true of time... not that we can stop in time, anyway...

That is how I would explain it to a non-mathematician.

Niel de Beaudrap
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    And "pi time" fails even more because it takes the decimal expansion of $\pi$ and shoehorns the digits unchanged into a mixed-base system (twelve, sixty, sixty, ten, ten, ten, ...) – hmakholm left over Monica Sep 09 '11 at 21:49
  • Yep, though it's all sort of a pretense unless you're standing precisely some integer multiple of 30 degrees east or west of Greenwich, and forget to observe daylight savings time. – Niel de Beaudrap Sep 09 '11 at 23:02
  • Sorry... so what is the answer? I asked a similar question on physics.se about if any object is "exactly" 1 meter long, and the answers were somewhat circular (or trivial, e.g. "a meter is defined to be the length of a certain object, so yes, there is one object that is one meter long") – The Chaz 2.0 Oct 14 '11 at 03:10
  • Assuming that time is continuous, the answer is "yes". But it is never pi time for long enough for you to notice or say that it is; it passes away after an instant. Pi time is not a time that you are ever practically at, but a point that you pass. Same thing for distances. – Niel de Beaudrap Oct 14 '11 at 12:14