Happy Birthday Euler

[zeitghost]

-1 = exp( i * PI ).

Well, who'd have thunk it?

Euler's identity - Wikipedia, the free encyclopedia

Euler's formula - Wikipedia, the free encyclopedia

Feck me, I'd forgotten all of that.

Long long ago & far far away, I knew stuff.

[d000hg]

Has to be one of the more beautiful equations although I still don't understand how it just so happens to be true, unless we view pi and e as reciprocal in a weird way i.e. it's just one constant not two.

[NotAllThere]

This is how it works. Consider the complex plane, with real numbers along the x-axis and imaginary numbers along the y-axis (aka an Argand Diagram). It turns out that a.exp(bi) where a and b are real numbers, is a complex number reached by moving a along the x-axis, then rotating anti-clockwise around the origin b radians. There are 2.PI radians in a circle, so PI radians is 180°. Therefore 1.exp(PI.i) is equivalent to starting at (1,0) and then rotating anti-clockwise through 180° - which gets you to (-1,0).

Euler's equation is usually quoted as 1 + e^(PI.i) = 0. In this way 7 fundamental objects of mathematics are in one equation: 1, +, e, PI, i, = and 0. Exponentiation is derivable from addition, so is not included as a fundamental object.

[DimPrawn]

Has to be one of the more beautiful equations although I still don't understand how it just so happens to be true.

Because god wanted it that way.

[NotAllThere]

Because god wanted it that way.

God invented the integers. All else is of man.

[d000hg]

This is how it works. Consider the complex plane, with real numbers along the x-axis and imaginary numbers along the y-axis (aka an Argand Diagram). It turns out that a.exp(bi) where a and b are real numbers, is a complex number reached by moving a along the x-axis, then rotating anti-clockwise around the origin b radians. There are 2.PI radians in a circle, so PI radians is 180°. Therefore 1.exp(PI.i) is equivalent to starting at (1,0) and then rotating anti-clockwise through 180° - which gets you to (-1,0).

Euler's equation is usually quoted as 1 + e^(PI.i) = 0. In this way 7 fundamental objects of mathematics are in one equation: 1, +, e, PI, i, = and 0. Exponentiation is derivable from addition, so is not included as a fundamental object.

Yes yes but it's still a bit lovely the 2 main irrational constants of our universe are in some way reflections of each other.

[proggy]

God invented the integers. All else is of man.

Nope integers are man made, most early civilizations only had numbers for 1 and 2 with others being many. Also God does not exist.

[d000hg]

Integers always existed, they just weren't discovered you clot.

[sasguru]

God invented the integers. All else is of man.

Parroting Kronecker's dictum without reflecting on how e and pi are so ubiqitious in nature, and hence would be "God's constants" explains why you have just read Spivak and not done the exercises* (wherein lies the meat).

*And if you claim to have done them and found them easy, I'll call you a liar.

PS if anyone is inspired to learn more on Euler try Dunham's "Euler:Master of us all".

[proggy]

Integers always existed, they just weren't discovered you clot.

They are an abstract notation invented by man, unless you can show me an actual integer? Maybe you should take a basic Math course before insulting people and making yourself look like a moron.