Favourite Mathematical Functions & Equations

[<pyridinyl_30>]

My favorites include the definition of the derivative:

f'(x)=lim h-->0 ((f(x+h)-f(x))/h)

and

lim x-->0 (sin x)/x = 1.

Could someone please explain e^(ix) = cos x + (i)sin x?

What situation does it physically apply to?

 

[Unknown]

x = -b +-sqrt b2 - 4ac / 2a

 

[johanneschimpo]

^ whoa, don't get all quadratic on us now!

 

[INFaMaS]

Speaking of which, what's the difference between arc connected and path connected.

I assume you know the definition of path connected. A space X is arc connected if any two points can be connected by a path f that is a homeomorphism between the unit interval [0,1] in R and f([0,1]) in X.

The easy counterexample to show path connected does not imply arc connected is the non-Hausdorff, partially ordered set [0,1] U 0', where 0 and 0' are incomparable and less than everything in (0,1]. Then it's easy to show that in this space, all points are path connected. (0 and 0' are path connected because you can go into the interval (0,1] and come back out to the other zero.) But 0 and 0' are not arc connected, because a path joining 0 and 0' will certainly not be a homomorphism. Suppose you have a path such that f(0) = 0 and f(1) = 0'. Well, 1 > 0 in R, but 0 and 0' are incomparable! So no arc exists and we have a path connected but not arc connected space.

 

[irishcoffee]

Could someone please explain e^(ix) = cos x + (i)sin x?

What situation does it physically apply to?

This particular equation is used a lot in signals analysis.
Converting a signal from the frequency domain to the time domain.

This formula can be interpreted as saying that the function eix traces out the unit circle in the complex number plane as x ranges through the real numbers. Here, x is the angle that a line connecting the origin with a point on the unit circle makes with the positive real axis, measured counter clockwise and in radians The formula is valid only if sin and cos take their arguments in radians rather than in degrees.
source: Wikipedia

For x=pi, the equation is
e^ipi = -1 or e^ipi + 1 = 0 or
e^ipi = 0 - 1 ; ln [e^ipi] = ln [0/1]; hence, ipi = ln [0] = 0 ;
e^0 = 1 ; 1 + 1 = 0 (mod 2)= 2

See 'Proof' by David Auburn(p.73-4)
"Let X equal the quantity of all quantities of X. Let X equal the cold ... months [11, 12, 1, 2] ... and four of heat [5, 6, 7, 8] leaving four months of in-/determine temperature [2,3,9,10].
[The months are a unit circle, and the Euler equation is its design. The indetermined area is undefined as is the definition of infinite.]
"Let X equal the month of full bookstores [infinite or undefined as month 10]. The number of books approaches infinity as the number of months of cold approaches four."

http://fermatslasttheorem.blogspot.com/2006/02/eulers-formula.html

 

[irishcoffee]

Favorite equation, I forgot:

Unsolvable:

for any x, y, z and n > 2

x^n + y ^ n = z^n

cannot be solved for x, y, z and n as whole integers.

Or something to this effect, can't remember what its called.

 

[monotome]

e^(π i) = -1
thst's the one I would go for. The proof of Fermats last conjecture (stated as a theorem at the time, but no proof so really a conjecture). The proof is 108 pages long, so a little too bulky.
As a mathamatical method, Godal numbering must rank as the most important system ever. I mean, every single mathamatical formula or function can be represented by a unique number. How powerful is THAT!

 

[Amberthefrog]

Roots of unity!

 

[INFaMaS]

As a mathamatical method, Godal numbering must rank as the most important system ever. I mean, every single mathamatical formula or function can be represented by a unique number. How powerful is THAT!

The coolest thing about Godel numbering is that you can assign unique natural numbers to the laws of natural numbers. Then you can prove results about the natural numbers that carry over to the axioms and properties of natural numbers which are represented by natural numbers. I believe Godel used this to help prove at least one of his incompleteness theorems.

 

[svacheme3]

My vote goes to Euler's Identity, e^(i*pi) + 1 = 0; and for some reason I always liked the integration rule u dv = uv - integral v*du, especially when I noticed I kind of could use it to find x*ln(x) - x as the integral of ln(x).

I also like integration rules that have infinite summations, like integral dx/ln(x) = ln(abs(ln(x)) + ln(x) + sum((ln(x)^k) / (k * k!)), k, 2, infinity)

No particular reason for liking either of these, just do.