## 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.

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Pythagoras’ theorem

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

$AC^2 =AB^2+BC^2$

Euler’s formula

$e^{i\pi}+1=0$, where $e$ is the Euler’s number.

Heron’s formula

$A=\sqrt{s(s-a)(s-b)(s-c)}$. where $A$ is the area of a triangle whose sides are of length $a,b,c$ and perimeter is $2s$.

Bayes theorem

$P(A|B)*P(B)= P(B|A)*P(A)$

Or

$P(A|B)=\frac{P(B|A)*P(A)}{P(B)}$

Sine rule
If A,B,C are vertices of a triangle, and sides a,b,c are a = BC, b = CA, c = AB then

$\frac{a}{sin(A)}=\frac{b}{sin(B)}=\frac{c}{sin(C)}$

Cayley – Hamilton theorem

every square matrix over a commutative ring (such as the real  or complex field) satisfies its own characteristic equation.

Euclid’s algorithm

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 = XYZ if
X = tan(A)
Y = tan(B)
Z = tan(C)

and $A+B+C = \pi$

Of course, this list is undeniably incomplete. There will be more entries, as I discover more gems.

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