History and applications
Paradoxes of the infinite
The ancient Greek philosophers appear to have been the first to contemplate the infinite in a formal way. The concept worried them somewhat, and Zeno came up with a number of paradoxes which they were not really able to explain properly. Here are two of them:
The dichotomy paradox
Suppose I travel from \(A\) to \(B\) along a straight line. In order to reach \(B\), I must first travel half the distance \(AB_1\) of \(AB\). But to reach \(B_1\) I must first travel half the distance \(AB_2\) of \(AB_1\), and so on ad infinitum. They then concluded that motion is impossible since, presumably, it is not possible to complete an infinite number of tasks.
The paradox of Achilles and the tortoise
A tortoise is racing against Achilles and is given a head start. Achilles is much faster than the tortoise, but in order to catch the tortoise he must reach the point \(P_1\) where the tortoise started, but in the meantime the tortoise has moved to a point \(P_2\) ahead of \(P_1\). Then when Achilles has reached \(P_2\) the tortoise has again moved ahead to \(P_3\). So on ad infinitum, and so even though Achilles is faster, he cannot catch the tortoise.
In both of these supposed paradoxes, the problem lies in the idea of adding up infinitely many quantities whose size becomes infinitely small.