Let's Start Over
The first question on your mind is probably, why start over? Can't we just continue where we left off? However at the end of Section II we saw that even though we can get very far using only an understanding of decimal notation, there is a point at which it becomes too complex to handle. We are going to need a much better grasp of the "process" in order to continue. To do this we have to retrace our steps, but with an eye for new and deeper insights. This is typical in the search for larger and larger numbers. More generally the process of mathematics is often like this.
So where did we go wrong, and how do we rethink numbers? Recall that in the beginning of Section I, the Counting Numbers were introduced by relating them to the concept of counting physical objects, such as apples, oranges, and the sheep in your flock. This was mainly to relate the concept of Numbers to something tangible and relatable. The problem with this is it gives the impression that numbers are as real as the apples, oranges and sheep, ... they are not. It also implies that if you have a number much larger than anything you can count, the number is somehow "less real" than say a number like 3, ... again, not true. Now open your mind to this frightening concept ... all numbers are equally unreal. Therefore there is no difference between 3, millillion, the square root of 2, i , e , Pi, or any other number you can come up with. What is important about numbers is not how they translate into physical reality ( for example, tell me what the square root of -1 would mean in reality.), but only how they relate to each other.
It is number relations that we are interested in here. We can construct an understanding of numbers that allows us to say 2 < 3 without having to make reference to sheep. To Begin let's consider Collections.
Number Sets
In mathematics the words "Collection" and "Set" are more or less synonymous. In both cases you have a set of "elements", which can in theory be anything, and they are all included within a "set". In our case, all sets will only contain numbers. You may have heard of the set of Natural Numbers, Whole Numbers, Integers, Rational Numbers, Real Numbers, Complex Numbers, etc. We will only need the set of Natural Numbers in our discussion of large numbers (for the most part). The simple explanation is that there are an infinite number of natural numbers, so that’s more than sufficient for creating extremely large finite numbers.
Usually math textbooks will introduce the Natural Numbers in the following informal way...
The set of Natural Numbers are the familiar numbers used in counting, starting with 1. Mathematically we denote this set by N, and it can be written as follows...
{1,2,3,...}
Where the dots means continuing on indefinitely.
The problem is this doesn't tell us much unless we already know something about numbers. But how do these concepts actually work in our minds? Do we have an infinite number of numbers floating in our head representing the set N ?
Of coarse not, but we know what infinity means, right?
At this point a distinction should be made between a finite set and an infinite set. A finite set can be counted, but an infinite set can not.
For example ... if I write {1,2,3} , this is a set with 3 elements, and they are the numbers 1,2, and 3. I have told you all of the elements. However if I write {1,2,3,...} and say there are more elements, have I explained to you what the set is? You might object, but it's obvious! Well sure, but hypothetically speaking if someone wanted to be pedantic they might tell you the next element is 5, then 8, then 13, and so on. You could then extend the set as {1,2,3,5,8,13,...}. Now how does this set continue ! Those with some familiarity with mathematics will, of coarse, recognize the fibonacci sequence. The basic trick is this ... the first two terms are given, namely 1 and 2. To find additional terms simply take the same of the previous two terms. So to find the third term we take the sum of 1 and 2. To find the fourth term the sum of 2 and 3. The fifth term would be the sum of 3 and 5, and the sixth the sum of 5 and 8. Now we can find the seventh term by taking the sum of 8 and 13, giving us 21. With this rule in hand we can of coarse continue with the 8th,9th,10th, ... as so on. However at any point in time we will only have a finite number of terms. We can never have the complete sequence.
So if no one knows the entire set at any point in time, does it exist? Mathematics asks us to believe that infinite sets really do exist in their entirety. For now, I ask you to at least consider this possibility.
Constructing Infinite Sets
Basically for any finite set, we can define it by simply listing all it elements. But we can not do this with an infinite set. Instead we have to define an infinite set by using some kind of rule. There are, generally speaking, two kinds of rules to define an infinite set.
1. Define the set by a conditional property
2. Define a set so that known elements generate new elements
To help you understand the distinction let's use the set of even numbers as an example. Let us call this set E. We can use a conditional property to define it. Let's say if n is a counting number, and is divisible by 2 without a remainder, then it's an element of E. We can write this as...
{ n | n is a counting number divisible by 2 without a remainder}
2 would be an element of n, because 2 divided by 2 gives us 1 R 0. So we would write ...
2 e E
We can also now write E as ...
E = { 2 , ... }
3 would not be an element of n , because 3 divided by 2 gives us 1 R 1. So we would write...
3 e! E
4 would be an element of n, since 4/2 = 2R0. So...
4 e E
and we can now write...
E = {2,4,...}
This is the process. Of coarse we know that we can get the set of even numbers simply by counting by 2's. This is basically accomplished by adding two each time to get the next element. Thus we can write ...
E = {2,4,6,8,10,12,14,16,18,20,22,24,26,28,30,32,34,36,38,40,...}
Without much effort. Mathematicians are fond of saying we can continue as far as we like. But as we learned in Section II, human beings can only count to somewhere in the hundred millions within their life time. Therefore there are definite limits to how many elements we can hope to write down. Despite this the condition definition of the set allows us to say that 2,000,000,000,000 is an element of E without having to count up to it.
We can also define the set of E using the 2nd type of rule. To do this we let one element “generate” another. We can write …
{n | n = 2 or n-2 e E}
This says “n such that n = 2 or n-2 is even”. Basically n is even if it is either = 2 or if n-2 is even. To see how this constructs the same set consider this.
2 is an element of E. Because 2 is an element of E, 4 is since 4-2 = 2 e E. Since 4 is now an element of E it follows that 6 is an element of E because 6-2 = 4 e E. This can be continued “indefinitely”.
I have developed another way to define a set. I call it Constructing a Set.
Constructing an Infinite Set
This is an important point of departure from the usual discussion of sets. For our purposes, we are going to treat infinite sets as if they are a work in progress. That is to say, we will agree to say the complete set exists hypothetically speaking, but it is strictly non-constructable. We can only generate larger and larger sub-sets of an infinite set.
This distinction is important for our discussion of large numbers because we really can’t know anything about large numbers until we attempt to generate them. I once said “If all the Natural Numbers exist, why is there this need to construct, define, and name large numbers?”. It’s fine to know there are an infinite number of them, but this tells us nothing about how to construct very large counting numbers. So we will now create a theory of Set Construction which will be handy in our later discussion of generating large numbers.
Our first assumption is that if S is an infinite set, we can not list all of it’s members explicitly. We can only generate a finite list of known elements of S. Call this list the set S’ (S prime). S’ must always be a finite subset of the infinite set S. (A subset is a set made up of some of the elements from a set).
Now assuming S is an ordered infinite set, there will be a first element, a second, a third, and so on. Let S’(1) be the set containing only the first element of S. Let S’(2) be the set containing the first two elements of S, and so on. For convenience we will say that S’(0) = {}. That is a set with no elements.
So if E is the set of even counting numbers, E’(1) = {2} , E’(2) = {2,4} , E’(3) = {2,4,6} , and so on.
Using this notation we can now define E in a third way.
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Constructing the Set of E
If E’(k) is known, Construct E’(k+1) as follows…
1. If E’(k) = {} then Let E'(k+1) = {2}
2. If E’(k) =! {} then Let E'(k+1) = E'(k) U {n+2} , where n = max[E'(k)]
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Although this may look somewhat intimidating to some, what it actually means is pretty self explanatory. Basically if we begin not knowing any even numbers, it tells us 2 is definitely an even number. To find additional even numbers, simply consider the largest even number you know an add 2. This will be the next even number in sequence.
Note that this is not that different than programming. This is basically the program you would write if you wanted a computer to construct the set of E.
In mathematics, a well defined step by step process is called an Algorithm. We will be using them a lot in the following chapters to generate large numbers.
Let’s begin with E’(0) and see how this algorithm constructs sequential sub-sets of E.
Firstly we note that E’(0) < E. That means it’s a subset of E. To “Construct” E’(1) we have to look at the following rules. Since E’(0) = {} we first let E’(1) = {}. Next we consider if E’(0) = {} then we add 2 as an element of E’(1). So E’(1) = {2}. Finally we consider if E’(0) =! {}. Since this is false we skip this step. Thus E’(1) = {2}
Now begin with E’(1) to construct E’(2). Since E’(1) < E we proceed to construct E’(2) by following the rules. First we let E’(2) = E’(1) = {2}. Next we consider whether E’(1) = {}. It does not, so we skip step 2. Step 3 asks if E’(1) =! {}, which is true. So we search through E’(1) for the largest element, 2 ( Note: Max[S’] takes the largest element of the finite set S’). It says to let 2+2 be an element of E’(2). Thus 4 e E’(2), and E’(2) = {2,4}.
Again E’(2) < E, and we know that the finite subset will always be less than E, so we let E’(3) = E’(2) = {2,4}. We skip step 2 since E’(2) =! {}, and perform step 3. We take the maximum of E’(2), which is 4, add 2, which is 6, and include it into the new subset E’(3). Thus E’(3) = {2,4,6}
This may seem like a lot of work just to count by twos, but this is actually good practice for understanding algorithms, which we are going to be using a lot of to construct large numbers.
As you can see, if rather tedious algorithm can be used to construct E. Simply by changing the initial element, and the rule for going from one element to the next, we can now describe a wide variety of infinite sets. For example, the Fibonacci sequence we saw earlier in this article can be defined as follows…
Let F be the Fibonacci set …
If F’(k) is a kth order subset of F, you can construct F’(k+1) as follows…
1. If F’(k) = {} then Let F'(k+1) = {1}
2. If F’(k) = {1} then Let F'(k+1) = F'(k) U {2}
3. If F’(k) =! {} or {1} then Let F’(k+1) = F'(k) U {n+m} , where n = max[F’(k)] & m = max(2)[F’(k)]
Note: max(2)[S’] is the 2nd largest element of S’. In general Let max(n)[S’] be the nth largest element of S’ if it exists.
Constructing N : The Infinite Loop
Now that we have laid down some more formal notations and notions we can define N (The set of ALL Natural Numbers). To do this we will use the concept of a super set. N is the super set of all N’(k). A Set S is a super set of S’ if every element of S’ is within S, but S contains elements which are not in S’. We will also use the concept of Union. Two sets can be combined into one using “U”.
For example {1,2} U {3,4} = {1,2,3,4}
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Definition of the Set of Natural Numbers, N
Let N >> A N’(k)
Let N’(0) = {}
If N’(k) is known, construct N’(k+1) as follows…
1. If N’(k) = {} then 1 e N’(k+1)
2. If N’(k) =! {} then N’(k+1) = N’(k) U {n + 1} , where n = max[N’(k)]
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Note that this definition doesn’t require us to know what N represents, only how to construct it. Unlike saying N = {1,2,3,..} this tells us how to find the next element in the sequence by simply looking over the sequence and applying the rules. But there is an important subtly hidden underneath all of the extraneous mathematical notation.
We don’t actually need to know what 1 represents, but in order to find the other elements we have to know how to add, naturally. That is a simple matter, so that may seem the least poignant thing in this discussion, but actually it is the most important point. Adding is in fact an algorithm in itself! However, how we choose to notate numbers isn’t actually important. For example, instead of writing N as {1,2,3,…} we could write it as { 1 , 1+1 , 1+1+1 , … }. In essence, the arithmetic is unnecessary. The only important feature of the natural numbers is the fact that it’s elements can be ordered and that given two unique elements, one must come after the other. But to do this we are going to have to come up with a theory of size order. We are also going to need to create some basic functions to relate numbers to each other.
In the next article I set up some basic axioms for the set N, and introduce what I call the 4 fundamental functions ( No these are not addition, subtraction, multiplication, and division! These 4 are even more fundamental than that! However you can use them to build certain operations, such as addition).
This may all seem not terribly relevant to large numbers at the moment, but I promise you once this new much stronger foundation is established we will be able to leave our previously huge numbers in the dust and be blasting off towards infinity in a hurry !
Stay tuned. I hope to get more work done on this site in 2010.