Eli5 Can someone tell me why some studies and mathematicians bother with the concept of infinite?

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“Bother”? It’s unavoidable.

The integers already form an infinite set.

The concept of limit, central to analysis, relies on it: what happens if we go past any bound, or make an infinite number of ever smaller steps? And that’s where the interesting things happen.

Finite objects can’t capture all of mathematics. For example finite-dimensional vector spaces (e. g. little arrows in the plane and such) are very nice but if you want to study functions from that point of view you have to do it in infinite-dimensional spaces. And again, you get new, interesting properties.

Even very classical geometry like the study of conics (circles, ellipses, parabolas and hyperbolas), which goes back to the Greeks, needs a line at infinity to become complete, removing exceptions you can’t get rid of without such an “ideal concept”.

Rest assured that with a bit of practice the infinite becomes quite familiar.

There’s a good documentary on this, A Trip to Infinity.

On the most basic level, we need the concept of infinity to define a circle. A circle is just a regular polygon with an infinite amount of faces.

Because it’s a very useful concept that allows us to formalize ideas about finite numbers that we’d really like to formalize.

It’s not that we *can’t* formalize things without introducing new symbols like infinity, but why would we? The whole point of math is to come up with symbols that abstract away the stuff you already did, so you can do it once and then never have to do it again. We have a symbol for the derivative because we don’t want to write lim(h->0) f(x+h) – f(x) / h every time, and similarly, we have a symbol for infinity (which usually represents a limiting process in one form or another – there are many different “infinities” in math) because we don’t want to write all that out either.