The basic idea of how to generalise "repeated multiplication" to the whole real continuum is to think about multiplication as a kind of scaling. Going from 2^n

The basic idea of how to generalise "repeated multiplication" to the whole real continuum is to think about multiplication as a kind of scaling. Going from 2^n to 2^(n+1) is ultimately about scaling the result up by a factor of x2. So, 2^(n+0.5) ought to be about scaling up halfway to x2, so that if you were to repeat that operation again, you got x2; and that's precisely sqrt(2). That reasoning gets you all the rationals. Since rationals are dense in the reals, impose continuity and you get all reals. This is the root of the algebraic definition of exponentials. But where there are reals there is calculus, and we also get a lot of insight from a differential definition. The key insight that leads to this definition is that if we could break down the exponential to 2^(n+epsilon) for a very small epsilon, we could stack up O(1/epsilon) of them to get wherever we like on the real continuum. So it makes sense that the definition dy/dx = a.y should produce the same function. This notion of "break down an operation into infinitesimal bits that stack up" can be taken seriously and formalised, and if you do that you end up with the theory of Lie groups and Lie algebras.