What is 10/3? If you enter that on your calculator, you will get the result 3.3… What does this mean? The ellipses means that the value after the decimal point and before the ellipses continue indefinately. But what does that mean? What is the value then? If you write 3.3 and add 100 more 3s, you would not have the answer. In fact, you can add any number of 3s – you will still not have the value because, as soon as you stop adding 3s and ask ‘Is this the value?’ you would have broken the ruleimplied by the ellipses – that of infinite repetition.
So in a very real sense, when you get an answer to a mathematical equation and the answer has an infinite decimal, then you have no answer at all – you simply have a ‘look, it is very close to 3.3…, but I can not give you an exact value’.
What I want people to realise is that the very best answer to the question “what is 10 / 3?”, is “10 / 3″. If this sounds absurd – consider that “10/3″ is simply a representation of a value. And it turns out, in fact, that 3.333… is also a representation – but a representation of an undeterminate value that only ever approaches the true value of 10/3.
Because we have 10 fingers, we are used to a base-10 mathematical system, and this system, unfortunately, pervades all of mathematics. The value “10″ is everywhere. The value 10 is implicitly in the value 0.5, for example. Why is 0.5 equal to “one half”? What does 5 have to do with halving? Because the first value after the decimal point stands for, literally, “divide this value by 10″. So 0.5 is exactly equivalent to (0*10) + (5/10). 10 is everywhere. This is a strength – and a weakness.
The weakness is this: obviously, not all values are divisable, without a remainder, into 10.
This means that our mathematical system is unable to accurately represent, in decimal form, many – actually an infinitely many – values. One of these values is the extremely simple concept “three parts out of ten”. It is a simple enough concept. If you have ten pebbles, and take three, then the ratio between the three compared to all ten is how many pebbles you hold in your hand compared to how many pebbles there are in total. Simply, it means, 3 out of 10! Duh!
And yet, in mathematics we have a drive to want to force this ratio – 10/3 – in terms of the decimal system and thus in terms of (y*1) + (x/10). I.e. – we think that the “real answer” is not “10/3″, but that it is (y=3) and (x=3) instead. We do this implicitly by writing 3.3… But herein lies the illogicalness (new word?) of it all – what we are taught to perceive as the more accurate / more acceptable answer, is in fact absurd, because it results in infinite regression, not only in the infinite repetition after the decimal point – but in the expanded, equivalent, equation form! And that is serious. An infinite equation does NOT equate any two determinite values, and is therefore not really an equation at all.
Note again that, by writing “3.33…”, and considering the meaning of the decimal point, this is simply shorthand for:
(3*1) + (3/10)
We write this as the answer to (10 / 3 = ?).
(3/10) and (10/3) are related. For any values x and y, (x/y) and (y/x) relate so that (x/y) * (y/x) = 1. This just means that as 10 divided by 3 can not be represented as a base-10 value, neither can 3 divided by 10 be represented as a base-10 value – and in fact, the ‘error’ produces when attempting to do so for 3/10 is a ‘power of -10′ of the error produced when attempting the same for 10/3.
Hence the equation itself is an infinite regression.
(10/3) = (3*1) + (3/10)
(3/10) = (0*1) + (3/10)
Do you see the infinite equational (new word?) regression?
That is my problem with the whole notion. The very notion is invalid, becasue if you were to try to solve this equation, you would never be able to arrive at a result – you CAN NOT arrive at a result because in order to get the result you have to get the result, but to do that you must get the result… the ’solution’ to the problem references the problem itself.
And this is the crux of the isue. It is like writing the problem:
x = x + 1, solve for x.
x can not be solved, because it exists on both sides in a such a way that it cannot be canceled out on only one side. The only concrete ‘fact’ we can show from this equation, is, brace yourself for a universal truth:
1 = 1
And x is nowhere in sight. For good reason – the laws of logic, equivalently the laws of mathematics, can simply not work with a flawed premise – a problem where the answer depends on the answer. Mathematics looks at absurd statements like that and says “BLEH.“
What is my point, you may ask? So what, you ask? I am reminded of friends and colleagues who ask of my artificial intelligence “But what does it do?” My point, dear reader, is this: pi is a logical fallacy.
