Calculus, differential equations, integrals, all rely on iterating towards infinite sums of infinitesimal values. It seems a bit dodgy at first but can be made

Calculus, differential equations, integrals, all rely on iterating towards infinite sums of infinitesimal values. It seems a bit dodgy at first but can be made rigourous by giving precise, consistent definitions of different infinite values and the operations between them. These are used extensively in many fields. This can be done by using limits "towards infinity / towards zero", but it can also be done more directly with precise defintions of infinities and infinitesimals: https://en.wikipedia.org/wiki/Nonstandard_calculus https://en.wikipedia.org/wiki/Nonstandard_calculus Infinite series, where an infinite sequence is summed (or multiplied, or ...). Many useful things fell out of reasoning about entire infinite sequences as if they are a single unit. I like this video about a method Newton found how to calculate pi efficiently, from playing with infinite series: https://www.youtube.com/watch?v=gMlf1ELvRzc https://www.youtube.com/watch?v=gMlf1ELvRzc Again it's possible to define everything about infinite series in terms of limits, but sometimes it's just a more useful thinking aid to think of infinite series as themselves. In a way, it's like dropping some unnecessary syntax sugar (the limits) as you realise you can still do useful things without it. And in fact it's possible to reason consistently about some kinds of divergent infinite series, where limits don't work but they are still logically consistent. Complex exponentials. The discovery and proof of Euler's formula, eⁱˣ=cos(x)+i⋅sin(x) comes out of reasoning about infinite series, showing that it's a consistent and useful definition of a complex exponential. Nowadays we use these exponentials to calculate all sorts of things in engineering and physics, we just take them for granted because they work and everything fits together consistently. Projective geometry, for example used in computer graphics. There is the concept of "point at infinity", which is actually represented by real values in a vector but in some respects you can think of it as representing "x/0" in the context of that geometry, and finding that further calculations using it still work out consistently to non-infinite results. Including the point at infinity as a value simplifies the system. Unlike with natural numbers where "infinity" is a complication that obeys different rules to numbers, the point at infinity in projective geometry removes edge cases and simplifies the rules. Elliptic curve cryptography also uses the "point at infinity" concept which arises from a geometrical or algebraic interpretation of the elliptic curve operations, and like in projective geometry it's a useful and consistent value to include, which is done because it simplifies the system to include it.