Pythagoras' constant will be referred to as the square root of 2. This is due to the following situation that arose which baffled many people and made some question the nature of physical property.
Consider this triangle. If we were to measure it's edges and said that it was perfectly 1, we could therefore conceive of making such a triangle in real life - at the very least, the vertical and horizontal vertices. This is perfect geommetry and has accuracy problems in real life. For example there is always a better accuracy with which to measure a distance - as we take our accuracy to it's limit we have a number closer and closer to 1. We just say it will converge to the number 1.
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While the thought is in my mind look at this trick :
m = 0.999999
10m = 0.999999 x 10
(10m - m) = (9.999999 - 0.999999) = 9 = 9m
9m / 9 = m = 1
So 0.999999 = 1?
Co-incidently, try it with 0.111111 and see what you get!
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Sometimes what Maths shows us is counter-intuitive, but one must consider one's position as to the meaning of life. Whether one would seek truth in the physical apparancy of the universe by measure and consistency or truth in within one's own perception, judgement or beliefs.
...To continue with the triangle, imagine starting by standing at the bottom right hand point. Then walking exactly 1 to the West. Now you are at the right-angle. Now move exactly 1 North to the top point of the triangle. In reality we have moved 2.
But what is the distance between the start and the finish - as the crow flies? Well using pythagoras' theorem, we have that one-squared add one-squared is 2. Square rooted it gives us that the distance between start and finish is the 'square root of two'. Which we call pythagoras' constant.
This number is an irrational number which means that the number can never be defined - it goes on forever. It has been defined to millions of places using an algorithmic process that gets more accurate every year. It can be seen below to a good accuracy:
1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799....
According to legend, Hippasus (the gentleman who discovered the irrational numbers) had made his discovery while out at sea, and was subsequently thrown overboard by his fellow Pythagoreans for having produced an element in the universe which denied the doctrine that all phenomena in the universe can be reduced to whole numbers and their ratios. Pythagoras would not accept the existence of irrational numbers although he could not deny that they exist.
This was also a crazy notion to the Egyptians who were amongst those baffled by the seemingly impossible task of physically measuring this distance. Interesting properties of the square root of two are as follows:
It can be used to approximate Pie...
Babylonian Reference
In YBC 7829, the Babylonian clay tablet (c. 1800 - 1600 BCE) gives an approximation of root 2. (See tablet above)
The Greeks had their fair share of ponderance upon the matter, take this conversation for eample:
Theaetetus: Theodorus was proving to us a certain thing about square roots, I mean the square roots of three square feet and five square feet, namely, that these roots are not commensurable in length the the foot-length, and he proceeded in this way, taking each case in turn up to the root of 17 square feet; at this point for some reason he stopped. Now it occurred to us, since the number of square roots appeared to be unlimited, to try to gather them into one class, by which we could henceforth describe all the roots.
Socrates: And did you find such a class?
Theaetetus: I think we did; but see if you agree.
Socrates: Speak on.
Theaetetus: We divided all numbers into two classes. The one, consisting of numbers that can be represented as the product of equal factors, we likened in shape to the square and called them square or equilateral numbers.
Socrates: And properly so.
Theaetetus: The numbers between these, among which are three and five and all that cannot be represented as the product of equal factors, but only as the product of a greater by a less or a less by a greater, and are therefore contained by greater and less sides, we likened to oblong shape and called oblong numbers.
Socrates: Excellent. And what after this?
Theaetetus: Such lines as form the sides of equilateral plane numbers we called lengths, and such as form the oblong numbers we called roots, because they are not commensurable with others in length, but only with the plane areas which they have the power to form. And similarly in the case of solids.
to be continued....
