If you ask people their opinion about Mathematics, their answers will most likely vary between absolute dread and sheer hatred. As always, I beg to differ. I like Maths - in fact I like them quite a lot. I was never exceptionally good at Maths, probably above-average though. In Physics we use them a lot, of course, but in a watered-down version, that is, we use all the nice results, without really bothering how they came about. This suits me quite well, because I could never adhere to the strict practice of proving everything. However, this does not mean that I don't enjoy the logic behind an elegant proof. And I say enjoy, because Maths (same as Physics) can actually be mind-bogglingly fun, if you take away a bored (and boring) high-school teacher and exam pressure.
You want a proof? Of course you do, it's Maths after all. So, let's play around with numbers and see some Fun Facts about them.
Mathematicians like to put numbers together in little collections called sets. Here I use the word "collections" instead of "groups" because the linguistic concept of a group is very, very different to the mathematical concept of a group (but that is another story). In Maths you should actually be very careful how you phrase things.
The most basic set of all is the set of natural numbers, often symbolised just with the letter N. These are of course our "normal" numbers, the ones we use for counting, that is N = {0,1,2,3,4,...}. The three dots mean that the set of natural numbers is infinite - it never ends, no matter how long you count. In other words, give me whichever natural number n and make it as large as you want. I can always give you a number that is n+1.
Our first Fun Fact deals with the natural numbers. As you probably know, we have two "types" of natural numbers. The odd ones (like 1,3,5,7 etc) and the even ones (like 0,2,4,6 etc).
FUN FACT 1 = there are as many even natural numbers as there are natural numbers in general.
Wait, WHAT???
How is that *even* possible (see what I did there)? Only every other natural number is even, so there must be half as many evens as there are naturals (or if you wish, twice as many naturals as there are evens). If you take the numbers from 1 to 10, for example, you only find five evens (2,4,6,8,10). So, clearly, there must be more naturals than there are evens.
Well, not quite... The problem here is that we are dealing with infinite sets. Our minds are unfortunately not immediately and directly capable of dealing with the concept of infinity and here is where the misunderstandings begin.
Naturally, we need to prove the Fun Fact. Don't stop reading just yet. It's actually not that difficult. For our proof, we need to introduce the mathematical concept of bijection.
You want a proof? Of course you do, it's Maths after all. So, let's play around with numbers and see some Fun Facts about them.
Mathematicians like to put numbers together in little collections called sets. Here I use the word "collections" instead of "groups" because the linguistic concept of a group is very, very different to the mathematical concept of a group (but that is another story). In Maths you should actually be very careful how you phrase things.
The most basic set of all is the set of natural numbers, often symbolised just with the letter N. These are of course our "normal" numbers, the ones we use for counting, that is N = {0,1,2,3,4,...}. The three dots mean that the set of natural numbers is infinite - it never ends, no matter how long you count. In other words, give me whichever natural number n and make it as large as you want. I can always give you a number that is n+1.
Our first Fun Fact deals with the natural numbers. As you probably know, we have two "types" of natural numbers. The odd ones (like 1,3,5,7 etc) and the even ones (like 0,2,4,6 etc).
FUN FACT 1 = there are as many even natural numbers as there are natural numbers in general.
Wait, WHAT???
How is that *even* possible (see what I did there)? Only every other natural number is even, so there must be half as many evens as there are naturals (or if you wish, twice as many naturals as there are evens). If you take the numbers from 1 to 10, for example, you only find five evens (2,4,6,8,10). So, clearly, there must be more naturals than there are evens.
Well, not quite... The problem here is that we are dealing with infinite sets. Our minds are unfortunately not immediately and directly capable of dealing with the concept of infinity and here is where the misunderstandings begin.
Naturally, we need to prove the Fun Fact. Don't stop reading just yet. It's actually not that difficult. For our proof, we need to introduce the mathematical concept of bijection.
"A bijection is a "rule" that introduces a *one-to-one correspondence* between two sets."
Bla, bla, bla, waaaaaaaaaaaay too technical... I sound like a bored (and boring) high-school teacher. Good, let's make it simple and have an example:
Imagine you have a room with people (that's one set) and chairs (that's a second set). You also possess the "rule", that is, you have the power to boss people around and tell them where to sit. A one-to-one correspondence means that each person in the room gets to sit on exactly one chair, and each chair in the room is occupied by exactly one person. Every person is sitting, every chair is occupied. Person <-> Chair. One-to-one.
Congratulations! You have found a bijection between the people and the chairs in the room. You can (hopefully) see that if you can find a bijection between two sets, it means that the sets have equal sizes; they have the same number of elements. In our example, there are as many people as there are chairs in the room.
We are all set now to prove the Fun Fact: take the set of naturals N={0,1,2,3,4,5,...} and take another set, let's call it S, consisting of all even naturals, S={0,2,4,6,8,10,...}. Now, take a "rule" n -> 2·n and apply it to the set N, i.e.
N -> rule -> S
--------------
0 -> 2·0 = 0
1 -> 2·1 = 2
2 -> 2·2 = 4
3 -> 2·3 = 6
4 -> 2·4 = 8
5 -> 2·5 = 10
...
As you can see, each number in N is transformed, via our "rule", to a number in S. In fact, each number in N is transformed to exactly one number in S, and vice versa, each number in S corresponds to exactly one number in N. We have one-to-one correspondence.
Imagine you have a room with people (that's one set) and chairs (that's a second set). You also possess the "rule", that is, you have the power to boss people around and tell them where to sit. A one-to-one correspondence means that each person in the room gets to sit on exactly one chair, and each chair in the room is occupied by exactly one person. Every person is sitting, every chair is occupied. Person <-> Chair. One-to-one.
Congratulations! You have found a bijection between the people and the chairs in the room. You can (hopefully) see that if you can find a bijection between two sets, it means that the sets have equal sizes; they have the same number of elements. In our example, there are as many people as there are chairs in the room.
We are all set now to prove the Fun Fact: take the set of naturals N={0,1,2,3,4,5,...} and take another set, let's call it S, consisting of all even naturals, S={0,2,4,6,8,10,...}. Now, take a "rule" n -> 2·n and apply it to the set N, i.e.
N -> rule -> S
--------------
0 -> 2·0 = 0
1 -> 2·1 = 2
2 -> 2·2 = 4
3 -> 2·3 = 6
4 -> 2·4 = 8
5 -> 2·5 = 10
...
As you can see, each number in N is transformed, via our "rule", to a number in S. In fact, each number in N is transformed to exactly one number in S, and vice versa, each number in S corresponds to exactly one number in N. We have one-to-one correspondence.
No matter which number n you choose from N, you can always find the unique, corresponding, even number s in S, through the rule n -> 2·n = s. And you can do this exactly because S is infinite.
So, our "rule" n -> 2·n is a bijection between N and S. But, as we have already seen, the existence of a bijection between N and S means that N and S have the same size. They have the same number of elements!
So, indeed, there are as many even natural numbers as there are natural numbers in general.
How is this for mind-boggling fun?
And a simple question to see if you have been paying attention:
Can you find a bijection between the natural numbers and the odd natural numbers? What does this mean?
