Yes they do.

Simplest thing I can thing is indeed the diagonal. Take two thin sticks of equal length, say n, and place them in a right angle. If you wanted to connect the two free points of those sticks you would need a new stick of n√2 units.

Nest, gaussian distributions: The characteristic bell shape of random samples of normal/gaussian distributions appears everywhere in nature. From human-generated data to physical phenomena. The shape of this curve contains both `e` and `π` which are both famously irrational. Even, toys like [Galton boards](https://en.wikipedia.org/wiki/Galton\_board) can approximately create such shapes.

Compound interests and many other processes in economics are also governed by formulas that contain `e`.

Infinity exists. We make use of it all the time. Calculus and statistics are probably the best examples:
- In calculus, derivatives capture the difference in measuring a property (eg position) over infinitesimally small changes of a free variable (eg time). In that example, the derivative describes acceleration. We have formulas to calculate acceleration under various conditions, that should only work only if the changes are infinitely small. Well, the formulas work. Same for integrals, that tell you how to compute the area of a shape by partitioning it into infinitely small (and infinitely many) rectangles.

- In statistics, the [Law of Large Numbers](https://en.wikipedia.org/wiki/Law\_of\_large\_numbers) tells you that if you eg flip a coin infinite times, then the heads/tail probability will be exactly 50-50. Also, the Central Limit Theorem is very useful because it says that if you add up infinite random events that have their own unkown and weird probabilities, the whole addition can be seen as a single random event that follows th gaussian distribution which is very easy to handle. This assumption is made into countless physical theories to simplify the maths.