Thursday, May 26, 2011

A trip down the number line

This was written by the user Hector_Lemans, 2009-09-12 09:43:17 PM

I've tried to track it down online, perhaps it was an article from somewhere else, but it looks like it's original. I'm cleaning out my Google notebook, and wanted to keep it here for posterity.

I've always been fascinated by large numbers. As a child, I would occasionally play the "one-up" game with my friends. Perhaps you've played this game. Two kids disagree about something and one says "bet you a hundred bucks it's not!" The other one retorts "bet you a million bucks it is!" And so on into infinity...which we usually got to rather quickly. But there are quite a few numbers between 100 and infinity. I'd like to take you, dear reader, on a trip to visit a handful of those numbers in the hope that you too can share in my wonderment of big numbers. So grab a power-bar, put on some running shoes, and let's begin.

Our journey will be entirely on the number line. As you may recall from grade school, the number line is a straight line with the counting numbers ticked off at regular intervals.

We start at point zero and begin moving straight ahead. We haven't worked out in a while so we're kind of slow off the starting blocks, but fairly quickly we pass 1, then 2, then 10, then 20. Now we're picking up some speed! As we pass 100, we notice that about a quarter of the numbers we just passed by were bold. As it turns out, we are on a special number line that has all the prime numbers in bold. Prime numbers are numbers that can only be evenly divided by 1 and themselves. So, for instance, 37 is a prime, because there is no other number we can divide it by (besides 1 and 37) that will give us another whole number. The fact that our number line bolds all of these primes is merely a curiosity...for now.

Fairly quickly, we pass 1,000; then 10,000; then 100,000. Now we pass the smallest of our big numbers: one million. A million dollars is what most people define as rich. Even a million pennies add up to $10,000 - not exactly chump change. You would need to flip a coin a million times before having a fair chance of getting 20 tails in a row. In other ways, though, one million isn't that big at all. One million seconds is just over 11 days. A million feet is only 190 miles - a few hours trip in the car. Ten good-sized novels contain a million words. Let's continue on.

We pass 5 million, 50 million, 500 million, and still we are gathering speed. By the time we hit one billion, barely 5% of the numbers we pass are bold prime numbers. A billion seconds is 32 years. A billion feet is 190 thousand miles - that's over three-quarters of the way to the moon! On the other hand, a large drop of blood has about a billion red blood cells in it. Several thousand public companies in the US have a market capitalization of over a billion dollars. So let's continue on in our search for big numbers.

50 billion, 500 billion, one trillion. Now we're approaching the high end of any numbers that come up in everyday conversation. The US debt stands at almost five trillion dollars. One trillion seconds is over 31,000 years - far older than recorded history and deep in the middle of the last ice age. One trillion feet is approximately the diameter of the earth's orbit around the sun. One light-year (the distance light travels in a year) is about six trillion miles. Alpha Centauri, our nearest neighboring star, is about 4.4 light-years distant. Going there in the space shuttle would take about 38,000 years. You would need to flip a coin at least a trillion times to have a fair chance of getting 40 tails in a row...but at 3 seconds a flip, that would take about 95,000 years. Think 3 seconds a flip is too fast? Take 5 seconds a flip and you're up to 160,000 years. On the other hand, one trillion drops of water is only a few hundred thousand gallons - a big pond. Most laptop computers can perform a trillion calculations in under half an hour. The average person is made up of about 50 trillion cells. Onward we go.

We pass 100 trillion and on to a quadrillion. It's hard to make out the numbers as they whiz past us, but a bit less than 3% - every 30th number or so - is a prime. That's interesting. Going from zero to a million, the concentration of primes dropped by over 95%, but going from a million to a quadrillion drops the concentration down only two more percent. Obviously, the primes are thinning out, but at a greatly decreasing rate. Will we soon pass by the last one? Let's find out.

We pass 100 quadrillion, then make our way into the quintillions. Our nearest neighboring galaxy, Andromeda, is about 15 quintillion miles away. We speed up a bit and find ourselves in the sextillions. At least, that is their official name. Another, more helpful way of visualizing these numbers is by scientific notation. Using this notation, one sextillion is 10^21. The little superscript 21 to the right of the 10 means we multiply the 10 by itself, 21 times. So 1,000 would be 10^3 (10 x 10 x 10), and 1,000,000 would be 10^6. Notice that the superscript also tells us how many zeros are in the number if written out in full.

We pass the sextillions by and move on into the septillions (10^24). From high school chemistry class, you may recall a unit of measurement called the mole. One mole of any element has the same number of atoms: 6 x 10^23, or just under one septillion. One mole of gold is a bit less than half a pound. One mole of hydrogen gas weighs only a gram. Coincidently, the visible universe is also about 6 x 10^23 miles across, or 93 billion light-years. The earth weighs (if you could weigh the earth!) 10 septillion pounds.

Fast and faster we go. The number line is a blur of gigantic numbers, with about every 100th one in bold. I say "about" because there doesn't seem to be any real pattern to how the prime numbers are spaced out - sometimes a couple primes will be within two or three numbers of each other, and other times there will be thousands of numbers between two consecutive primes. No matter, onward we must go.

In no time at all, we are deep in the octillions (10^27). The human body contains approximately 7 x 10^27 atoms. If we divide that by the 50 trillion cells we estimated earlier, we find the average human cell contains over 100 trillion atoms. Think about that - each cell in your body is constructed of more atoms than there are bricks in the Great Wall of China!

We put on a burst of speed and race past the next several large numbers. Nonillion (10^30), decillion (10^33)...the names don't mean much anymore. It's easier to just go by the scientific notation. We move on past 10^50 - approximately how many times you'd have to flip a coin to have a good shot at getting 150 tails in a row. As we move up to 10^70 and beyond, we start approaching the limits of anything physical these gigantic numbers can be compared to. The known universe has 10^80 atoms (give or take). If we assume the universe is a sphere, 93 billion light-years in diameter, that amounts to less than one atom per cubic foot! We live in a universe of mostly empty space.

Huffing and puffing, we finally make it to a truly, incredible, gi-normous number: a googol. A one followed by a hundred zeros - 10^100. The name "googol" was coined by a professor's nine-year-old nephew when asked to think up a name for a really big number. Have we run out of primes yet? Not hardly - less than half a percent of the numbers we run across are primes, but that still works out to 1 out of every 230, on average. Now, take a deep breath and scarf down that power bar. To reach our last few big numbers, we're going to have to sprint.

We blow past a googol and make our way up to 10^124. This is the number of particles you'd have if the universe were nothing but a solid mass of electrons - no empty space at all. We run past 10^155 and bump into some numbers used every day by millions of people! Notice how random those bold prime numbers always appear to be, no matter how far across the number line we go? Turns out, that randomness makes them perfect for encoding sensitive data - like a credit card number - and transmitting that information over the Internet. The actual process is quite complicated (as you might expect), but the essential steps are as follows: Two very large prime numbers are chosen and kept secret by the bank. These two primes are then multiplied together to form a huge composite number. This composite number is given to the person with the credit card, and is used to encode the credit card information, which is then sent back to the bank. The bank, having the original two primes the composite number came from, factors them out and retrieves the data. The beauty of the system is that the encoding using the huge composite number is a one-way process if you don't know the primes it is made of - factoring composite numbers with upwards of 155 digits is next to impossible, even with the fastest computers. So it doesn't matter if someone figures out the composite number you're using, they still can't decrypt an encoded message.

From here on out, most of the interesting numbers we'll run across will be prime numbers. Mathematicians like to find and collect huge primes, like some sort of super nerdy baseball card collector. The late Samuel Yates collected what he called Titanic primes - prime numbers with more than 1,000 digits. Chris Caldwell, of the University of Tennessee, took up the torch after Yates' death in 1991 and stores a list of these large primes on the Internet, with a new category for prime numbers with more than 5,000 digits called Gigantic primes. Several of these primes are quite strange. One, that has 5,114 digits, is composed of only 1s and 0s. Another, with 6,400 digits, is all 9s except for one 8. Currently, the largest known prime, discovered on September 4th, 2006 is just shy of ten million digits long. As we reach it on the number line, we marvel at its size. Written in exponential notation (similar to scientific notation but the base number doesn't have to be ten), the current largest prime number is 2^32,582,657 - 1.

And now, let us accelerate to a speed only possible in our mind's eye. Numbers blur together, numbers so big that just writing them down in full would be a problem in the real world...not enough paper! On and on we go. There it is, just up ahead - the number that will end our journey. We slow down from our impossibly fast pace and stop on the number line directly over this stupendous number. A googolplex - a one followed by a googol zeros. If the whole universe were nothing but paper and ink, you still couldn't even write down the number in full (never mind trying to imagine how big it is). This is the end of our journey, but we can look ahead and see even bigger numbers...a tiny select few are even in bold. We still haven't run out of primes! Do the primes ever end? Does this number line ever end? Does infinity really exist? Just because we can say a thing doesn't make it real or even possible - I can describe a universe in which 2 + 2 = 5 or a square is round, but that doesn't mean a place like that could exist...does it? If we squint our eyes and look really hard, we can see a number so large that no number appears to follow it. It's a number so large we can't even describe it in words. Is that the last number? Only one way to find out. Let's go...last one there is a rotten egg!