Big numbers, Google or googol?

This article is mostly about big numbers – Some really large ones too. A googol (where Google got their name from) is the scientific label for a number 10 to the power of 100. Or 10^{100}. Or a 1, with 100 zeroes after it. It’s a very large number indeed. Archimedes once calculated the number of grains of sand in the universe to be 10^{63}. Not even a Googol. So why might we need such large numbers, after all we have light years as a measurement for long distances? One reason we need such large numbers to be defined is to describe how much data is manipulated every day on the computers of the world.

Starting with my own machine, I estimate that at a clock speed of 2.9 GHz, my CPU can deal with a linear 2.9 x 10^{9} bytes per second of 64 bit binary chunks– that does not include the parallel processing that goes on inside a modern CPU. With all the other 25 or so chips on my motherboard, which run at 400Mhz, that’s 400 Million (4 x 10^{8}) bytes per second, in 32 bit pieces.

So my motherboard + CPU performs (64 x 2.9 x 10^{9})+ (25x32x4x10^{8}) =1.856×10^{11} + 3.2 x10^{11}

That is 500,000,000,000 binary data bits every second. In an 8 hour day that is 8x60x60x5x10^{11}

We are still a little behind the Archimedes total grains of sand in the universe figure too. The earth is made up of 10^{50} atoms, so that’s getting closer. The universe contains conservatively estimated 10^{79} hydrogen atoms – that is a big number. How about the volume of the universe measured in cubic millimeters? At 13.7 billion light years in diameter, the volume of the calculable universe is 5 x 10^{31 }Km^{3}, which is 5 x 10^{40 }m^{3 }= 5×10^{49} Cm^{3 }= 5×10^{58} cubic millimeters.That comes to 1.44 x10^{16} x 365 = 5.25 x 10^{18 }bits per year,and that is just my computer.

Globally, there are over 1 billion computers, so the daily tally, just for computers switched on for 8 hours a day is 5.25 x 10^{18 }x 10^{9} = 5.25 x 10^{27}. Imagine all the other technology such as digital TV and cellphone usage and we soon reach vast numbers of bits, a guess of 10^{40} seems to be on conservative side, but we are still not at a Googol.

This is not helping at all, we are still not at a Googol yet.

So what is the point of a Googol? – “Google” the answer and you won’t find much, so the short answer is “not a lot apparently”; A Googol (not Google) has no particular use in mathematics, and in physics can describe the number of subatomic particles in the universe and maybe provide the opportunity to calculate things on a near-infinite scale. Handy stuff for the cosmology dept. of your local university for sure. For the rest of us, not so useful.

So the only other contenders in the real world for such large numbers are games; apparently the game of chess has 10^{123} potential moves but the world record holder is the game Arimaa which has 10^{402} gameplay combinations.

Next time you Google Googol, you might also consider some other numbers that are still very impressive and within our mental horizons. Google handles 3 Billion searches per day from over a billion users. 46% of their searches are for products or purchasable services. 20% of searches are for local business. There are 152 million blogs online (this is one of them) and over a billion regular readers. 100 million of them in the US. One third of all US consumers spend over 3 hours per day online. So although the figures are not quite a googol in magnitude, they are impressive. They indicate that business is shifting more and more toward online dependency. Millions of web pages are added every day, thousands of websites appear to soak up that content and existing sites expand at a frightening rate. Some article directories have upwards of 10 million pages – Ezine alone has over 400,000 contributors.

The growth of the internet and Google (or Googol) search volumes are not exponential, but they are growing. With an increasing number of users, hardware interfaces and uses, the internet is rapidly expanding and like the universe, it seems like there will be no end to it. When will we reach a time when a Googol (not Google) is in daily use as an internet statistic? Hmmm, might be another trillion or so years at current rates.

Still, in the meantime, you can spend a moment to consider what we might use the largest of all recognized numbers. Google is big; a googol is gi-normous. Just what in the universe do we need a Millinillion for? For those who aren’t as smart as a 12^{th} grader, that’s 10^{3003}. A googolplex is a googol^{googol} which is an incomprehensibly large number. To even write down that number would take more space than the universe contains. Mathematicians constantly seek to prove the unproven and some of that requires even larger numbers. The largest known representation of a number is Grahams Number. That cannot even be written down in scientific notation nor in a power tower . This must be the limit surely? What is it used for?…some weird theorem that most of us care little about and even fewer can understand.

To find out more about numbers and algebra here’s some academic input from those who consider these things every day!