Different Sizes of Infinity

Before you read this make sure you have a basic knowledge of sets.

The "cardinality" of something is the size of a set. For example ­the cardinality of the set {3, 2, 17, -9} is 4, and the cardinality of the set {2, 6, -7} is 3. But when thinking it terms of infinity things become more confusing, I mean how can you categorise different sizes of infinity, infinity surely must just be infinity, right?

Well unfortunately, no. Using the ideas of sets we can find that there are actually different cardinalities of infinity. But to find this we need a more general definition of what being the "same sized set" is. A set {1, 2, 3, ..., x-1, x} has a cardinality of x and the set {1, 2, 3, ..., y-1, y} has a cardinality of y, the sets will be the same size if x = y. This then leads to the definition that being the "same size" is if the sets have the same amount of elements.

When we check if a set is the same size as another set, we simply count the number of elements in both sets and compare. But with infinite sets this is (obviously), impossible. But if we break the task down a bit we find ways of managing this. Take the elements from each set and pair them up, if they do match up perfectly with no doubling up or ignoring any elements then the sets are the same size.

Now if we pair the natural numbers, N {1, 2, 3, ...}, and the integers, Z {-2, -1, 0, 1, 2, ...}, we will be able to see which set is larger. We pair the numbers up like so:

And you can see, that despite your better knowledge, the numbers are pairing up and will continue to do perfectly infinitely. This then means that the sets of both the natural numbers and integers are the same size and thus have the same cardinality. This cardinality is referred to as: ℵ₀ (read as "aleph null" or "aleph zero"). The aleph, "ℵ", character is used as it is the first letter in the Hebrew word 'infinity'. ℵ₀ refers to 'countable infinity', or in simpler terms the lowest possible form of infinity.

When we continue to pair up the numbers we discover that the rational numbers and the natural numbers are too the exact same size and thus the cardinality of the rational numbers is also ℵ₀. In fact the prime numbers are also deemed to have the same cardinality as the natural numbers, integers and rational numbers. I know it seems insane, but it is true.

And going back to my earlier point there are larger cardinalities of infinity. As you would think the next cardinality of infinity is ℵ₁. The size of the real numbers in fact are larger that the natural numbers, integers, etc., this is because the real numbers (in now way at all) can be paired to the natural numbers. Because, between 0 and 1 there an infinite number of real values that are there, in fact the same can be said between 0 and 0.000000001. Thinking of it in this way helps to visualise the fact that the set of real numbers must be larger than the set of natural numbers.

As far as ℵ_x goes it will exist as long as x is a natural number, but as x goes past 1 it becomes much harder to visualise how this infinity is larger than the last. But there can not be ℵ_1.5 for example, this is the continuum hypothesis.

Modulus

You may or may not be aware of what modulus is. If you have come across the modulus (absolute value) of a number before it is likely to only be of real numbers. The notation for the modulus of x is, |x|.

If the number is just a real number then calculating the modulus is incredibly easy and requires no thought. Essentially it is just 'taking the positive value' of the number, for example: |5| is simply 5, and |-3| = 3. It follows that any rational or irrational real number is just the positive value of it. The definition of |x| for any real value of x therefore follows that |x| = √(x²).

Another, potentially easier way to think of the modulus of a number, is to think of the numbers distance from 0 on a number line. -5 is 5 units away from 0 and 5 is also 5 units away from 0. This thought is useful when it comes to finding the modulus of a complex number.

Complex numbers can also be represented (in a way) on a number line, they are represented in a two-dimensional complex plane. A complex number, 5+3i will be 5 units in the x direction and 3 units in the y direction, therefore the general form of a complex number is x+iy.

As I said previously, to find the modulus of a complex number it helps if we think of our previous definition of |x| being the distance from 0 on a number line (or a complex plane, in this case). If we think of it in this manner then simple co-ordinate geometry states that the distance from 0 to x+iy is √[(x-0)²+(y-0)²], or simply the general definition of |x+iy| = √(x²+y²).

|x| = √(x²) when x is a real number, and this able to be proved from our new definition of |x+iy|. |x+iy| = √(x²+y²), and when x+iy is a real number, y has to equal 0, plugging this into the equation we get |x+i0| = √(x²+0²), therefore |x| = √(x²), proved.

If any of this was explained poorly or just went straight over your head please comment and I will do everything I can to make it understandable. Also if you would like more information on imaginary numbers please visit an older post of mine here.

Squaring Numbers Under 100

First, I must state that although you do not necessarily require any prior knowledge other than to be competent at maths, it does not hurt if you know the square numbers up to 10 and are reasonably good at quadratic equations.

I recently discovered this technique of timesing numbers together recently and I was pretty proud of myself and have been using to bamboozle my friends and family. However I did not create this technique, I do not know who did, but if you do please leave a comment.

I am starting by squaring a number, as the skills learnt from that will help you when you times two different numbers together, which I will cover in a later blog post.

To begin with when squaring a number we need to choose one number as a base, we will call it 'a', for example if we had 43 our base would be 40. If we had 47 however I would personally choose 50, as dealing with 3 is easier than dealing with 7. This second number we will call, 'n', to get 'n' you take the base from the original number, 'o'.

n = o-a

You may see where we are going here, we will take the equation (a+n) and square it, that is all we have to do. The product of this will be a²+2an+n². So for example if we wanted to find the answer to 76²:

a = 80, n = 76-80 = -4
80²+2(80*-4)+(-4)²
6400-640+16 = 5776

After a little bit of practice you can do this very efficiently and very quickly in your head.

I am alive!

Clearly I haven't been posting nearly as much as I should have. Things have unfortunately got in the way, what with my GCSE exams, starting college etc. But now I am back, well, for now. A fair few of my points may be on contents of AS Maths and Further Maths, but there is likely to be many mathematical tricks and the hidden gems of maths in there too. Anyway, I just thought I would update you will on my current situation and my hopeful resurgence to the world of blogging.