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Large and Small CHAPTER 14 LARGE AND SMALL As they used to say on Monty Python: ‘Now for something completely different.’1 The challenge: Finding the shortest way to write enormous and tiny numbers. Squeezing a large number into a small space in written or graphic form is something that people have been trying to do for thousands of years. The ways we do this today aren’t the same as the ways that were used in the past. MEET THE PROBLEM SOLVERS It’s a sunny spring Sunday afternoon in New Jersey in the 1920s. Three people are taking a walk: a couple of boisterous boys, Milton and Edwin Sirotta the former around nine years old, the other slightly younger; and their uncle, Edward Kasner, a happy-go-lucky professor of mathematics at Columbia University. The professor invented a game for the boys to play to see who can spot the extreme examples of things; the largest tree and the smallest one; the littlest insect the biggest one. Then the professor suddenly said, ‘All the things you’ve spotted, you’ve been able to name, but I’ve got an enormous number in my head – a one with a hundred zeros after it, but I don’t know what to call it … it doesn’t have a name. What do you think? Can come up with a name for it?’ Some sources report that Kasner asked Milton to come up with a word for ‘a word with a sound that had a lot of ‘0’s in it’ Milton thought for a while and then coined the name “googol”2 which can be written like this: 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. It was first published in a book co-written by Kasner and James Newman, Mathematics, and the Imagination.3 There are several theories as to why Milton came up with that name. At that time, there was a very popular American comic strip called ‘Barney Google and Snuffy Smith’ created by cartoonist Billy DeBeck in 1919.4 When Milton Sirotta was challenged to propose an even larger number than a googol by his uncle, he suggested a number “one, followed by writing zeros until you got tired.” Kasner wasn’t satisfied, pointing out that “different people get tired at different times, and it would never do to have Carnera a better mathematician than Dr. Einstein, simply because he had more endurance.”5 (Primo Carnera was an Italian boxer who held the heavyweight title for a year before running into Joe Louis, the World Heavyweight Champion from 1937 to 1949.6) 217 CHAPTER 14 So Kasner defined a googolplex to be one followed by a googol of zeros.7 that certainly would tire out everyone. According to Kasner, it’s so long a number that ‘there would not be enough room to write it, if you went to the farthest star, touring all the nebulae and putting down zeros every inch of the way.’8 Astronomer, Carl Sagan “estimated that writing out a googolplex in standard form (i.e. 10,000,000,000…) would be physically impossible since doing so would require more space than is available in the known universe.9 Is there a briefer means of representing a googolplex and other large numbers? Well, if there isn’t, this entry has come to an abrupt end. THE METHOD Think about it. In base 10, units are arranged in groups of 10 units, organized from 0–9, 10–99, 100–999, etc. Any large number can be written in base 10 with a digit between 1 and nine at the start (it won’t have a 0 at the beginning). Being in base ten, large numbers can be reduced by dividing them by ten to get down to a digit between 1 and 10, followed by a fractional amount, where relevant. And that works in reverse. What number do you need to multiply ten by to get to 100? How many 10s did you need to use? How many zeros in 100? What about if you want to get to 1,000? How many 10s did you use? How many zeros in 1,000? What’s the pattern? Any small number can be written in base 10 with a ‘0.’ and be followed by digits between 0 and 9 (the number will, by definition, be smaller than 1), or it can be written as a fraction with a one on top as the numerator. How much do you need to divide one by to get to a hundredth, or 0.01? How many 10s did you need to use? How many places did the decimal point move? In which direction? What about a thousandth, or 0.001? How many 10s did you use? What’s the pattern? THE NUMBERS A googol is a number larger than the largest numbers used in physics and astronomy.10 It can be denoted by10100. The googolplex can be denoted by 10googol The sun’s mass is 1,989,000,000,000,000,000,000,000,000,000,000 grams. “That’s 330,000 times the mass of the Earth.”11 Really? No, that’s an approximation. How accurate? That’s a subject for another time. Too bad Kasner didn’t ask his nephew to coin a name for some small number, say 0. 00000000 (a hundred zeros), then a 1. It could be written 1/googol and an even smaller number would be 1/googolplex. A very small number is the mass of an electron: 0.000000000000000000000000 00000091096 grams, another approximation. “The mass of a proton is 2000 times the mass of an electron.”12 Writing very large and very small numbers isn’t the only thing that’s tedious. Saying their names is even worse. Quickly can you say the name of the number 12,345,678,909,876,543,210? Do you want to try to name it? Thought 218.
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