 epi or pie – how to tell which is bigger without a calculator?

Short answer: eπ is bigger, because (smaller number)(bigger number) > (bigger number)(smaller number) when those numbers are greater than e.

Long answer: The numbers e and π are about 2.71 and 3.14. If you want to figure out which is bigger (eπ or πe) you might first try just rounding down to an integer and ask which is bigger: 23 or 32? They’re pretty close – 8 and 9 respectively – but if we include the decimals it looks too close to call.

Instead, let’s study a more general relation and see if we can’t apply what we learn to the problem at hand:

xy = yx

This is a transcendental equation, meaning that we can’t solve it for a simple closed form function of x. We need to use tricks, like guess-and-check or plotting.

First, notice that there is a trivial set of solutions when x=y. But more interestingly, there is a class of solutions that make another line that looks sort of like 1/x and is satisfied by points like 24 = 42 (in fact, you can prove (2,4) and (4,2) are the only integer solutions to this equation quite easily). Lastly, that curve trends towards x=1 when y gets big, and vice versa. For example, the point (1.0495, 100) approximately satisfies this function. it makes for a final plot that looks like: xy = yx

Notice that the two curves intersect right between 2 and 3 – that intersection point is (e,e)! Calculus tends to do things like that. Furthermore, we can use this plot as a sort of map using the plotted line as boundaries. For example, off of the plotted lines we enter ‘inequality territory’ and for a given x and y either xy is bigger or yx is bigger. We only need to check one point in each region and we know the behavior there – I’ve labeled them I-IV. The points I’ve chosen are (3,5) and (1,2) because they’re both easy to calculate and easy to find on the map. Straight forwardly, 21 > 12 and 35 > 53 (243>125), so we know that xy > yx in regions I and III while xy < yx in regions II and IV.

Finally, we can apply this to the problem at hand. Which is bigger, eπ or πe? It’s straight forward enough to just check which region of the plot this point is in. So the point (e,π) falls squarely in region I, where xy > yx, so we know that  eπ > πe! In fact, this plot gives us an interesting rule. Thanks to the symmetry of regions I and II, we get a general rule for when any pair of x and y are bother greater than e:

(Smaller number)(Bigger number) > (Bigger number)(Smaller number).

In fact, by going below e on the plot, you can get a similar rule in regions III and IV but with the inequality flipped!

image credit: Wikimedia Commons

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