October 2004

# When Google becomes ePlay

When the folks at Google were preparing for their IPO earlier this year, one of the facts they had to give was the revenue they expected the initial sale of stock to generate. Their answer: \$2,718,281,828. In the event, that figure turned out to be unrealistically high, but how come they were able to be so precise?

MAA Online readers should have no difficulty seeing the insider joke here. Estimating a stock valuation of around \$2 billion, the company's geeky founders (both former computer science graduate students at Stanford) took the digits from the decimal expansion of e, the base for natural logarithms.

Mathematical constants are nothing new to Google. Its very name is a derivation of the word googol, a term invented in 1938 by nine-year-old Milton Sirotta to denote a 1 followed by 100 zeros. Milton was the nephew of the American mathematician Edward Kasner, who introduced the concept as a throwaway example of an extremely large number in his book Mathematics and the Imagination. A googol is greater than the number of particles in the known universe, which is estimated to be between 10^72 and 10^87. Even bigger than a googol, according to Kasner, is a googolplex, a 1 followed by a googol zeros.

Before the rapidly growing Google moved to larger premises in its Silicon Valley home earlier this year, its old headquarters was called the Googleplex, and the mathematical constants e, pi, and i were used to number buildings. When the company was planning its new quarters, the new location was temporarily called the i-plex, before being renamed the new Googleplex when they moved in.

This summer Google turned to the number e once again for assistance, this time to attract potential employees. Large billboard suddenly appeared in Silicon Valley and Cambridge, Massachussetts, bearing nothing other than the legend:

"(first 10 digit prime in consecutive digits of e).com"
[Click here to see a photo of the billboard alongside Silicon Valley's Highway 101. And click here to listen to an NPR report on the ad's appearance.]

Although doubtless meaningless to most passersby, this ad was red rag to a bull for the math-minded techie types that it was aimed at. A straightforward computer search quickly reveals the first 10 digit prime number in the decimal expansion of e to be 7427466391, so I am not spoiling the fun by giving the answer here. If you then visit the website www.7427466391.com, as the billboard ad instructed, you will be presented with a more difficult puzzle. Solve that and you will be taken to a web page that asks you for your CV.

All in all, it makes you wonder why the Bay Area search engine company did not call itself e-Play. No, wait, that name is awfully reminiscent of another successful Silicon Valley startup.

While on the topic of e and other mathematical constants, I can't resist repeating the claim that I've made to generations of math students that I've taught over the years that my absolute favorite mathematical equation of all time is the one discovered by the great Swiss mathematician Leonhard Euler in 1748, that connects the five most significant and most ubiquitous constants in mathematics:

e^(i.pi) + 1 = 0
To me, this equation is the mathematical analogue of Leonardo Da Vinci's Mona Lisa painting or Michaelangelo's statue of David. It shows that at the supreme level of abstraction where mathematical ideas may be found, seemingly different concepts sometimes turn out to have surprisingly intimate connections. Consider:

The number 1, that most concrete of numbers, is the beginning of counting, the basis of all commerce, engineering, science, and music. The number 0 began life as a mere place holder in computation, a marker for something that is absent, but eventually gained acceptance as a symbol for the ultimate abstraction: nothingness. As 1 is to counting and 0 to arithmetic, pi is to geometry, the measure of that most perfectly symmetrical of shapes, the circle - though like an eager young debutante, pi has a habit of showing up in the most unexpected of places. As for e, to lift her veil you need to plunge into the depths of calculus - humankind's most successful attempt to grapple with the infinite. And i, that most mysterious square root of -1, surely nothing in mathematics could seem further removed from the familiar world around us.

Five different numbers, with different origins, built on very different mental conceptions, invented to address very different issues. And yet all come together in one glorious, intricate equation, each playing with perfect pitch to blend and bind together to form a single whole that is far greater than any of the parts. A perfect mathematical composition.

Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence. It brings together mental abstractions having their origins in very different aspects of our lives, reminding us once again that things that connect and bind together are ultimately more important, more valuable, and more beautiful than things that separate.

Devlin's Angle is updated at the beginning of each month.
Mathematician Keith Devlin ( devlin@csli.stanford.edu) is the Executive Director of the Center for the Study of Language and Information at Stanford University and The Math Guy on NPR's Weekend Edition. Devlin's newest book, THE MATH INSTINCT: Why You're a Mathematical Genius (along with Lobsters, Birds, Cats, and Dogs), will be published next spring by Thunder's Mouth Press.