The most beautiful equation ever

It is said that one day, when the great German mathematician Leonhard Euler was at the Court of Catherine the Great, a friend asked him for a proof of the existence of God. Euler is supposed to have replied with what is now called the Euler Identity and then essentially said Q.E.D.

Adam Frank explains the Identity and detail and points out just why it’s both profoundly surprising, and to a numbers and relationship sort of person, beautiful, in a post over on the NPR Cosmos and Culture blog:

“The best equation ever – the incomparable and glorious Euler’s Identity.  Here is it:

ei(π)+1 = 0

Why are so many mathematically inclined folks sent into paroxysms of delight over this string of symbols which seem like gibberish to others.  Let me quote from Paul Nahin’s wonderful Dr. Euler’s Fabulous Formula: Cures Many Mathematical Ills –

I think [Euler’s Identity] is beautiful because it is true even in the face of enormous constraint.  The equability is precise: the left hand is not ‘almost’ or ‘pretty near’ or ‘just about’ zero, but exactly zero.  That five numbers each with vastly different origins, and each with roles in mathematics that can not be exaggerated, should be connected by such a simple relationship, is just stunning.  It is beautiful.  And unlike the physics or chemistry or engineering of today which will surely appear archaic to technicians of the future, Euler’s equation will still appear, to the arbitrarily advanced mathematicians ten thousand years hence, to be beautiful and stunning and untarnished by time.

So to begin with, its the appearance of those five magic numbers that starts things off.  Lets go down the list.

  • Zero (0) which can be added to any number and the number is unchanged
  • One (1) which can be multiplied to any number and the number is unchanged
  • Pi which appears everywhere from trigonometry on to the most abstract domains of mathematics
  • e which is the base of natural logarithms.  It is also a number which ‘just shows up’ again and again in the exploration of mathematics
  • i which is the fundamental imaginary number.  There is no number which when multiplied by itself gives a negative result i.e. 1*1 = 1; (-1)*(-1)=1.  So someone just invented a new number: i = Square Root(-1).  They called it imaginary (cause it is) and so much mathematical magic suddenly appeared as to almost defy imagination.  How can an imaginary number be so central to mathematics?”

Read the full article here.

Seems like a profitable reflection for Labor Day.

Author: Nick Knisely

Episcopal bishop, dad, astronomer, erstwhile dancer...

1 thought on “The most beautiful equation ever”

  1. Interesting post and interesting blog all around for sure! My only comment to the 5 special numbers is I could make it 6 by multiplying each side by say… 15. Then 7 by dividing by say…7. Then….
    I admit it is cool such an equation cancels, but I am still not sure why the 5 numbers are special other than they are just what is required to get the job done. I will have to think about this more deeply.

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