The set of all numbers is infinite, and there are numbers as large as we want. What to do with these illustrious gentlemen? Does the world need such big numbers?
One way to see that the set of all numbers is infinite is to assume the opposite. Let’s imagine that the set of all numbers is finite. If so, there would be a number greater than all the others. Let’s add 1 to this colleague. The result will be a number that wasn’t in the all-number drawer (simply because it’s bigger than their “biggest” number). Now, this is absurd! And then? When we suppose that the set of numbers is finite, we get a wrong conclusion. So the premise is wrong, and there are infinite numbers.
A consequence of this argument is that there are numbers as large as we want. But here’s the beauty of math: we don’t have to turn to bigger and bigger numbers to see that they are infinite. There are infinite numbers between 0 and 1. The average between 0 and 1 is 1/2. The average between 0 and 1/2 is 1/4. Repeating this calculation about ten times, we arrive at 1/1024. And continuing indefinitely, we get infinite numbers between 0 and 1 – smaller and smaller. It’s a simple exercise, but there’s something wonderful about it when done for the first time. Let’s go back to the very large numbers.
It is estimated that the human body has around 31 trillion cells; in scientific notation it is written 3.1 x 1013 cells. The number of stars in the universe is estimated at around 50 sestillion, or 5 x 10.22. And the number of atoms in the observable universe would be something like 100 bi-billion, or 1 x 1080. And the most interesting question: why use numbers greater than the number of atoms in the universe? Is there anything so spectacular as to justify the use of numbers greater than the universal quantity of the smallest unit of things?
Well, of course the math guarantees us the affirmative answer. Let’s look at the number Pi. We know that Pi is a number close to 3; so what would it have to do with large numbers? Simple: Pi has infinite decimal places! There are more decimal places in Pi than there are stars in the sky. Or atoms in the universe. It’s eternity after the comma. Getting as many decimal places out of Pi as possible is getting to know an important ingredient for science in general better and better. Whether to estimate the diameter of the Milky Way with the precision of a hydrogen atom, or to prepare us as a civilization for the challenges that lie ahead. A curiosity: the sequence of decimal places in Pi is infinite and not repeated; therefore, it is possible to find the day of birth of anyone there.
Although (very) tempting, relating Pi to very large numbers can be cheating. After all, any liter of gasoline costs more than two Pi’s. Let’s build up courage and face very big numbers!
A few centuries ago, mathematicians Marin Mersenne and Pierre de Fermat used to write letters to each other. In one of them, Mersenne would have asked Fermat to factor the number 100895598169 – that is, to find numbers whose multiplication results in 100895598169, on the condition that these numbers are prime. Fermat kept the conversation going smoothly and replied: the number suggested by Mersenne could be written as the product of the primes 112303 and 898423. At first glance, these are not small numbers. But when it comes to cousins, things are more subtle.
Currently, a relevant computational effort is dedicated to finding new prime numbers, as there are an infinite amount of them out there. The most recent cousin discovered would not fit in this post. It’s number 282589933-1, which has 24862048 digits. That is, there are approximately five times as many digits in this number as there are letters in an edition of the bible, in any language.
In scientific notation it is written as 1.4 x 1024862046! There are therefore many more digits in the most recent known prime number than there are atoms in the universe or stars in the sky. And? In theory at least, knowing ever-larger prime numbers can be useful for getting personal data — like credit card numbers — around safely.
Be it gigantic or tiny numbers, or even the distance between one prime number and the next: if Hamlet had taken a course in number theory, he might have told Horace that there is much more to mathematics than there is between heaven and earth.
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Edgard Pimentel is a researcher at the Center for Mathematics at the University of Coimbra and a professor at PUC-Rio.
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