# Is it ever really Pi Time?

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?

• 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 9 '11 at 21:11
• 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 9 '11 at 21:12
• 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 9 '11 at 21:18
• @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 9 '11 at 21:40
• And when is hammertime? – The Chaz 2.0 Sep 9 '11 at 21:55

• 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, ...) – Henning Makholm Sep 9 '11 at 21:49