# e^{pi} or pi^{e} – how to tell which is bigger without a calculator?

**Posted on:**November 13, 2015 /

**Categories: Uncategorized**

**Short answer:** e^{π} 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 (e^{π} or π^{e}) you might first try just rounding down to an integer and ask which is bigger: 2^{3} or 3^{2}? 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:

### x^{y} = y^{x}

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 2^{4} = 4^{2} (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:

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 x^{y} is bigger or y^{x} 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, 2^{1} > 1^{2} and 3^{5} > 5^{3} (243>125), so we know that x^{y} > y^{x} in regions I and III while x^{y} < y^{x} in regions II and IV.

Finally, we can apply this to the problem at hand. Which is bigger, e^{π} or π^{e}? 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 x^{y} > y^{x}, so we know that e^{π} > π^{e}! 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

Have a question? Send it to matt@quarksandcoffee.com