My eleven year old son has been fascinated with large numbers lately. He’s asking all the typical thought provoking questions that precocious eleven year olds tend to ask, like “how many stars are there in the universe?” and “how long would it take to walk to Alpha Centauri?”
And of course, the number Googol is always a great benchmark, as in “Are there more than a Googol water molecules in the ocean?”
So, how big is a Googol? I came up with the following example to explain to him how big a Googol really is. (And yes, the number Googol (10 to the 100th power) is spelled differently from the search engine, Google).
In scientific notation, a Googol is typically displayed as 1×10100. Written out, this is a “one” with 100 “zeros” behind it, as such:
10,000,000,000,000,000,000,000,000,000,000,000,000,000, 000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
One interesting thing about our ability to use symbolic notation to represent and manipulate numbers is that we really don’t have to think about all the intermediate numbers when performing math problems. For example, we can add 2,500 and 1,200 in our heads and come up with a sum of 3,700. That’s easy. And in doing so, we don’t need to think of – or even be aware of – the number 2,942, for example (or any of the other numbers in between). If we had started from 2,500 and added 1,200 by counting, we would have had to go through each integer number, so that every number in between touches our consciousness – if even for the briefest instant. (And I specifically say “integer” to limit the exercise to whole numbers – we’re not even going to touch on the subject of irrational numbers!)
This ability to use symbolic notation to represent and easily manipulate quantitative numbers is very powerful, and a big time saver – but it can be a bit of a crutch, as it allows us to compute VERY big numbers without having to really think about their absolute magnitude. Counting from one up to any number definitely gives us an appreciation for the size of that number.
Can we count to a Googol? It will take some time ...
Let’s compare this to the age of the universe. By all current scientific estimates, the universe is roughly 13.75 Billion years old (1.375×1010), or when written out:
Age of Universe (in years): 13,750,000,000
Wow! The age of the universe (in years) is much smaller than a Googol.
How about if we counted one number every second? Each year has about 31.5576 million seconds, so multiply 13,750,000,000 (years) by 31,557,600 (seconds in a year) and you get:
Age of Universe (in Seconds): 433,917,000,000,000,000
It’s a big number, but you haven’t even scratched the surface of trying to count to a Googol. If you counted one number every second, starting from the Big Bang, you have only gotten up to 433.917 quadrillion.
Current “state of the art” personal computers can add numbers together roughly at a rate of a billion times a second. (That’s not entirely true – but for the sake of this exercise, let’s pretend that we have a computer that counts numbers at a rate of a billion times a second). If our computer started counting at the moment of the Big Bang and added one number a billion times a second, the count as of today would be roughly:
Age of Universe (in Billionths of a second): 433,917,000,000,000,000,000,000,000
So a modern day desktop computer could not count to a Googol in the time that the universe has existed. Clustered “super computers” are much faster, but that’s not going to buy us much. Even if a super computer were a million times faster than our desktop PC, you would take that last number and add six more zeros – which still isn’t anywhere near a Googol.
But our fastest super computers today surely can’t be the fastest computers possible, could they? There must be ways to compute faster – whether it’s humans or some other intelligent life, we obviously don’t know the top computing speed possible. There is, however, an absolute physical limit to how fast any computing device can go.
Max Planck, founder of modern quantum physics, defined the absolute smallest “quantized unit” of time, space, electrical current, and temperature which exists in our universe. “Planck Time” is basically the amount of time that it takes a photon to travel one “Planck Length” (at the speed of light). It does not matter how advanced humans (or aliens) might be, no one will ever be able to construct a computer that could add numbers together faster than once per Planck Time (and in reality, probably not faster than once every few billion Planck Time units). What is the duration of Planck Time? It’s 5.39124×10−44 seconds. That’s the shortest time period you can ever have.
Ok, so we take the reciprocal of that – one divided by Planck Time – and we find an absolute physical limit of 1.854860×1043 calculations per second. Multiply this by the number of seconds since the big bang, and our “fastest possible computer” which started counting at the moment of the Big Bang has now counted up to ...
8,048,556,547,287,822,467,558,483,762,548,100,000,000,000,000,000,000,000,000,000
Or put another way, 8.04855×1060. Whoo-hooo! Yeah! We’re up to 1/1.2424×1039th of the way to a Googol!
Work these numbers backwards, and you’ll find that a Googol “Planck Time Units” is ... about 1.7083×1049 years. This is how old the universe will have to be for our “fastest possible” computer to count to a Googol.
Remember that 1050 is NOT half of 10100 – not even close. Half of a Googol would be 5×1099. So if you have a line that is a mile long, and you want to split it into a Googol equal units, 1099 units would be just a tenth of a mile! (1099 multiplied by ten is 10100.) 1050 would not even be width of an electron.
So, do large numbers like Googol have any “real world” application? For most people in every day life, no ... not really. It is estimated that the total number of sub-atomic particles in the ENTIRE UNIVERSE is somewhere on the order of a Googol – so cosmologists and other physicists may work with numbers like this on occasion.
However, prime numbers greater than a Googol are used in cryptology and security. Currently, the largest known prime number is 243112609 – 1 , a number which has 12,978,189 digits (and which also happens to be a “Mersenne Prime” because it can be expressed in the form of 2n – 1). If we can’t count to Googol (which only has 100 digits) in the time that the universe has been in existence, then you can see why it would be hard to break a security key which is based on a number with over 12 million digits!
Working with even larger numbers, like the ever popular “Googol-plex” (a digit “one” with a Googol zeros behind it) are fun mental exercises, but not really useful even for scientific or engineering purposes. Carl Sagan pointed out that writing out a Googol-plex in the smallest readable font wouldn’t even fit inside the known universe.
So we’ve definitely answered “No” to the question of “are there more than a Googol water molecules in the ocean”. Now, to walk from Earth to Alpha Centauri ... where’s my calculator ...
(Note: Since this post was originally written, a larger prime number was found in October 2014: 257,885,161 − 1, which is another Mersenne Prime.)