# Mathematics

The Cali Garmo does Math

Tag: Infinity

## Primes – How high can we go?

Jan 5

When we last left off I had finished talking partially about prime numbers and how to find them. But some of you may have been wondering, how many primes there are in total? Is there some way to maybe find a formula to find a prime number?

If you look at the primes, they seem to go on forever! And thanks to amazing mathematicians from the past we know that they do! But HOW do we know that they actually do? How can you be sure there is a number after 97 that is prime? Some will just point to their computer and say, “Computers can prove that!” and they would be partially right. Computers have helped us find extraordinarily HUGE prime numbers. For example, remember those special prime numbers called Mersenne primes which are primes of the form  where p is also prime? Well, a worldwide collaboration was started in order to find the largest prime known to man and they called themselves GIMPS.

The organization is called GIMPS or Great Internet Mersenne Prime Search. You can find their website here: [GIMPS] Quite recently they found the largest prime ever known to man. What is the prime number you may be wondering? It is: . You might be thinking that this looks like a small number, but it really isn’t. This prime is 17,425,170 digits long. That is HUGE! It is so big that if you were to print out the number on paper (without commas) it would be 4,283 pages long!

It took years for the computers to calculate that prime number, so as we questioned earlier, how do we know if there is a prime number greater than this or not? It is so large we can barely use computers to go any higher, and we can forget about writing the number down (you try writing out 4,283 pages of numbers!). So then how can we KNOW if there are infinite number of primes or not? Well this is exactly what a guy named Euclid was pondering over 2,000 years ago.  Euclid was able to prove that there are an infinite number of primes! So no matter how big of a prime number we find, there will always be a bigger one. So cool! But how did Euclid come up with this?

Below is a rough transcript of how Euclid created his proof, but before we go over the proof here a lemma that Euclid uses in order to prove it that you should know:
Lemma 1: Every number greater than 1 can be broken down into multiples of prime numbers. e.g: . Notice that prime numbers just equal themselves, and composite numbers are multiples of primes!

Euclid’s Proof: Suppose that there are only a finite number of primes. This means that there is a number x such that any number greater than x is not prime. All these numbers greater than x thus must be composite and by the lemma we just discussed must be a combination of prime numbers. So if we take every prime number less than x and multiply them together we get a number n. Now if we add 1 to n we get a prime number! (Notice that since 2 is a prime, thus multiplying all the primes together would get us an even number, which would automatically not be prime. So we add 1 to make it an odd number.) Why/how is n+1 prime?! Because no matter which prime number we select and we divide n + 1 by it we will get a remainder of 1. And since we have used every prime number less than x, and every number between x and n are composite, there are no other prime numbers that (n+1) can divide into. So (n+1) must be prime! But, we had stated earlier that x is the greatest prime and since (n+1)>x we see that our earlier thought was wrong (that there are a finite number of primes) and so there must be an infinite number of primes!

[Note: This type of proof is called proof by contradiction. I’ll eventually lay the groundwork for how this works, but for now you can look it up in google if you’d like.]

Wow, that was complex, but so rewarding! We can now for sure state that there are an infinite number of primes! Fun part is, that this was only 1 of many proofs that there are infinite primes! Any questions? Leave a comment!

Questionably yours,
The Cali Garmo

## Infinity

Dec 7

Say you have an infinite number of camels walking through the desert and a little baby camel is born, how many camels are there now? Surely there are still an infinite number, and surely since we added 1 more camel this infinity is greater than the original infinity. But is it even possible for infinity to be greater than infinity? So confusing!

The biggest error most people make when talking about infinity is to think of infinity as a number. In reality, infinity is not a number, it is a concept. By definition infinity is a quantity without end. So if you take a number and you keep adding 1 to it, then you are going toward infinity, but you never actually hit infinity since infinity is not a number.

So since infinity is not a number, what happens when you ‘add 1’ to infinity. As we saw earlier if we keep adding one we still have an infinite amount. So by adding 1 we are still at infinity. Now looking at the number of camels, even though we added 1 camel to the group, we still have an infinite number of camels and so the ‘number of camels’ has not changed. It’s a weird concept to grasp, but this concept (created by Georg Cantor) helps us understand infinity.

So then we are naturally inclined to ask, can we ever make a certain infinity greater than another infinity? Weird concept to think about, but thanks to Cantor and a bunch of other super smart mathematicians, we now know the answer is yes!

This concept is created in Set Theory where we look at the number of elements in a set. Cardinality is just the number of elements in a set, and a set is just a collection of objects. So if you have a group of 10 people, then the cardinality of the group is 10. The cardinality of the number of months is 12, the cardinality of the number of weeks is 52, the cardinality of the set of birthdays is 366. Well how about the cardinality of a set like the natural numbers? The natural numbers are the numbers 1, 2, 3, 4… So what is the cardinality since it seems like there is no end to the number of items in the set.

This is where the concept of infinity comes into play. Since the number of items in the set of natural numbers never ends, there are an infinite number of them. We call this type of infinity, countably infinite. The reason we call it countable is because we as humans can sit there and count them. We may never be able to hit the end (since there is none), but we can count them. (We can first say 1, then say 2, then say 3, etc. And if you say 1 number per second for 70 years you’d hit 2,209,032,000 !)

Well are there more countably infinite sets? Yes, integers and rational numbers are also countable! We can count integers by counting them in this way: 0, 1, -1, 2, -2, 3, -3, etc. How about rational numbers?! There seems to be a crazy amount of rational numbers, and in between 0 and 1 there are an infinite number, so they must not be countably infinte, surely. But they are!

How did mathematicians find this out? They took the rational numbers (which are all integers and fractions) and put them into a grid like the following:

    …    …    … … … … …

Now in order to make it countable just go in diagonals. So our sequence would be: . Then in order to get all the negative numbers, just switch in between the two (positive, then the same number negative like we did for the integers.)

Now how about the set of all numbers (aka real numbers, which is the rational numbers with the irrational numbers such as ) It turns out that the real numbers are actually not countable. We can see this because there is no way to sit and count the real numbers. For example let us start with the number 1. What number do we choose next? 1.01 is wrong, because 1.001 is less and is also a real number. But we can’t choose that since 1.0001 is also a real number, so no matter what number we choose there is no logical next number to choose since we can keep finding a smaller number closer to 1.

Wait, now we have 2 different types of infinity! Countable and uncountable infinities. The cool thing is that if you look further into set theory you’ll find out that uncountable infinity has a higher cardinality than countable infinity! So there are the same number of whole numbers as rational numbers, but there are more real numbers than any other type of number. So cool!

If you want further information on Set Theory to be able to see different types of infinity I recommend A transition to advanced mathematics by Douglas Smith, Maurice Eggen, & Richard St. Andre (Books/Cole 2001) Have fun counting!