02 Apr The most beautiful formulae/theorems/identities in mathematics
This is my personal collection of formulae/theorems which I consider lovely. By “lovely”, I mean objects which possess a certain degree of Elegance and Simplicity. The formulae/theorems are listed in no particular order.
The most popular and fascinating theorem in Euclidean geometry takes the first place in the list.
If AB, BC and AC are three sides of a right angled triangle ABC, where AC is the hypotenuse, then
, where is the Euler’s number.
. where is the area of a triangle whose sides are of length and perimeter is .
If A,B,C are vertices of a triangle, and sides a,b,c are a = BC, b = CA, c = AB then
Cayley – Hamilton theorem
every square matrix over a commutative ring (such as the real or complex field) satisfies its own characteristic equation.
If a and b are integers and a > b, then gcd(a, b) = gcd(a (mod b), b)
Trigonometric gem 1
sin(x – y) sin(x + y) = (sin(x) – sin(y)) (sin(x) + sin(y))
Trigonometric gem 2
X+Y+Z = X*Y*Z if
X = tan(A)
Y = tan(B)
Z = tan(C)
Of course, this list is undeniably incomplete. There will be more entries, as I discover more gems.