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.

Image Source: Shutterstock

Image Source: Shutterstock

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^{ipi}+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 = X*Y*Z 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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S. Parthasarathy

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2 Comments
  • Taha
    Posted at 08:34h, 19 August

    I know you could have added more, but you have to mention the quadratic formula, it has unbelievable impact on the world of maths and it shouldn't go unnoticed.

    • Gonit Sora
      Posted at 22:30h, 27 August

      We agree with you, but it was the choice of the author. Thanks for the comment.