3 ms·

>“Five different numbers, with different origins, built on very different mental conceptions, invented to address very different issues. And yet all come togeth

>“Five different numbers, with different origins, built on very different mental conceptions, invented to address very different issues. And yet all come together in one glorious, intricate equation, each playing with perfect pitch to blend and bind together to form a single whole that is far greater than any of the parts. A perfect mathematical composition.” Well that's the thing. They're not unconnected. The hype depends on the people who aren't aware of the connection.

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 ago
  Add 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 ago
    I 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.
    - roenxi 5y ago
      "its own derivative" is defined in terms of things happening at infinity though. Calculus is very much a study of what happens to things at the infinite and infinitesimal; and e has its heft because of calculus.
      - eutectic 5y ago
        OK, agreed.
- huachimingo 5y ago
  Its 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.
simiones 5y ago
Aren't they historically unconnected, as the quote alludes to? 1's origin is in counting. pi's origin is in the study of the circle. 0's origin is in accounting. i's origin is polynomial algebra. e's origin is in the study of exponentials and derivatives. The relationships between all of these fields is a discovery in itself, and a revelation that was not apparent at all throughout history, and it is still not apparent to most people who haven't studied math in-depth. Basically, e^ipi + 1 = 0 is proof of the connection, one that is not at all immediately apparent, and has not been known for the majority of humanity's history.

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 ago
Add 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 ago
  I 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.
  - roenxi 5y ago
    "its own derivative" is defined in terms of things happening at infinity though. Calculus is very much a study of what happens to things at the infinite and infinitesimal; and e has its heft because of calculus.
    - eutectic 5y ago
      OK, agreed.
huachimingo 5y ago
Its 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.

simiones 5y ago

Aren't they historically unconnected, as the quote alludes to? 1's origin is in counting. pi's origin is in the study of the circle. 0's origin is in accounting. i's origin is polynomial algebra. e's origin is in the study of exponentials and derivatives. The relationships between all of these fields is a discovery in itself, and a revelation that was not apparent at all throughout history, and it is still not apparent to most people who haven't studied math in-depth. Basically, e^ipi + 1 = 0 is proof of the connection, one that is not at all immediately apparent, and has not been known for the majority of humanity's history.