4 ms·
It's most certainly not obvious how the relation df(x)/dx = f(x), the ratio between the diameter and circumference of a circle , and the square root of -1 -- a
by SimplyUnknown 5y ago
It's most certainly not obvious how the relation df(x)/dx = f(x), the ratio between the diameter and circumference of a circle , and the square root of -1 -- a "trick" to extend the number line with a second dimension -- are related, let alone be used to create an useful construct.
- roenxi 5y agoAdd in the fact that the reason they are related appears to involve infinity (power series). It is tough to argue against it being an impressive result.
- eutectic 5y agoI wouldn't say power series are the 'reason' they are related. The more intuitive reason for Euler's identity is the combination of e^x being its own derivative, and the relationship between complex multiplication and rotation.
- huachimingo 5y agoIts interesting to see the discrete case. Let D_n be a difference operator like D_n(F) = F(x+n) - F(x) Solving by induction for the equation D_n(F)/D_n(x) = F(x), F(0)=1, it gives (1+n)^(x/n). So now you have a "family" of functions that when "derived" by its difference, gives the same constant ^ x. Euler is the case when n→0. This aplies for every n, including complex numbers.