Irrational Magic

Fibonacci 

Some times I cannot distinguish between mathematics and magic. For example when I look at equations which containing irrational and rational numbers. For me it is pure magic that irrational numbers having infinite decimal expansion some how yield a rational numbers.

Irrational numbers: An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q. Irrational numbers have decimal expansions that neither terminate nor become periodic.

Φ2 – φ – 1 = 0

Here φ is the all beautiful golden ratio equal to 1.61803398874989484………… but when you square and subtract φ and 1 you get zero. All those problems with infinite series vanish. Another example is the about the most beautiful euler’s relation.

ei∏ + 1 = 0

e is an irrational number, so is ∏ and i is imaginary number, but they play a intricate role to give beautiful equation.

Previous post on beautiful equation.

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5 Responses to “Irrational Magic”


  1. 1 David Woodford April 19, 2009 at 5:33 pm

    also root 2 times root 2 gives 2.

    Root 2 is irrational and 2 is rational – this is true for many square roots

  2. 2 S.Karthikeyan August 17, 2006 at 2:22 pm

    Hi Jimw.

    Thanks for stopping by. Let us write in two equations one for -1 and another for 0. As noted this one stands out as the most beautiful equation ever scripted.

  3. 3 jimw August 17, 2006 at 2:19 pm

    Ooops! Of course you have to choose between -1 and zero in that relation – you can’t have them both. Must think before posting… ;-P

  4. 4 jimw August 17, 2006 at 2:19 pm

    Don’t forget zero and minus one! At one time both zero and negative numbers were seen as crazy abstract ideas and had to be assimilated slowly into the mathematical cannon. So Euler’s equation links all the major classes of ‘weird’ or ‘special’ numbers in the most unspeakable beatiful way. It really is the best equation in the world.

  5. 5 Srinivasa Ramanujam August 16, 2006 at 7:01 pm

    Its a real beauty that two irrational numbers e and pi are related together with an imaginary number in the classical euler’s equation.


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