Identity of Ramanujan (Elementary Algebra)

In elementary algebra is identity of Ramanujan a simple formula , which consists of the binomial and the rules for multiplying out of parenthetical expressions seen. It is attributed to the Indian mathematician Srinivasa Ramanujan , who recorded this formula in his famous notebooks . The Ramanujan identity can also be interpreted as the theorem of triangular geometry .

Formulation of identity

The equation always applies to two real numbers ${\displaystyle a,b}$

${\displaystyle (a+b-{\sqrt {a^{2}+b^{2}}})^{2}=2\cdot ({\sqrt {a^{2}+b^{2}}}-a)\cdot ({\sqrt {a^{2}+b^{2}}}-b)}$   .

Geometric interpretation

A right-angled triangle with the hypotenuse and with and as cathetus is given in the Euclidean plane . ${\displaystyle ABC}$ ${\displaystyle [AB]}$ ${\displaystyle [BC]}$ ${\displaystyle [CA]}$

On both short sides are removed with a compass so that in three stretch broken will, with between and lies and between and . ${\displaystyle [AB]}$ ${\displaystyle [AB]}$ ${\displaystyle [AR],[RS],[SB]}$ ${\displaystyle R}$ ${\displaystyle A}$ ${\displaystyle S}$ ${\displaystyle S}$ ${\displaystyle R}$ ${\displaystyle B}$

Then the equation applies:

${\displaystyle {\overline {RS}}^{2}=2\cdot {\overline {AR}}\cdot {\overline {SB}}}$   .

That means:

The square of the length of the middle section is equal to twice the product of the lengths of the two outer sections and . ${\displaystyle [RS]}$ ${\displaystyle [AR]}$ ${\displaystyle [SB]}$

In other words:

If you erect the square over the middle section and at the same time form a rectangle whose base sides correspond in length to the two outer sections, the area of the square is twice the area of ​​the rectangle.

swell

• Alexander Ostermann , Gerhard Wanner : Geometry by Its History (=  Undergraduate Texts in Mathematics. Readings in Mathematics ). Springer Verlag , Heidelberg, New York, Dordrecht, London 2012, ISBN 978-3-642-29162-3 , pp. 69 , doi : 10.1007 / 978-3-642-29163-0 . MR2918594

Individual evidence

1. ↑ Alexander Ostermann, Gerhard Wanner: Geometry by Its History. 2012, pp. 179, 369