I have always been fascinated by what are known as irrationals numbers (numbers that cannot be expressed as a ratio or fraction). Now, before you leap on me, I know ratios and fractions are different conceptualizations.
The ancient Greeks, called the Pythagoreans (named after their leader Pythagoras), were entranced by whole numbers - unlike many kids in schools today.
For the Greeks, the whole numbers were 1, 2, 3, 4, 5, and so on ..... Zero had no been invented yet.
They also had the concept that there could be fractions made up of "ratios" of whole numbers 1/2, 3/4 ...
They reasoned quite reasonably, that whole numbers and fractions covered everything.
However one day, so the tale goes, on a boat, one Pythagorean demonstrated that you could not get a number (as fraction) multiplied by itself to result in 2.
You could get close (7/5) x (7/5) = 49/25 (1.96) and (10/7)x(10/7) = 100/49 (2.041) ...
However, you never never get there (a/b)*(a/b) can never ever equal 2
It was something that did not appeal the the Pythags or either to mathematicians up to fairly recent times. Some suggested that these irrational numebrs had suspect ontological status.
So they said √2 , √3, √5, √6 and infinitely many more …
Could not really be real numbers ...
Think of the number line. It seems reasonable, doesn't it dear reader, to assume that you can keep subdividing it up using fractions until you fill it completely.
Just take the gap between 0 and 1. Surely we can pepper it with fractions. 1/2, 1/4, 1/8, 1/16, 99/100, 999/1000 and so on and so forth.
And using fractions we can always put another point between two points ad-infinitum. So that this ............ eventually becomes ________ !!!
So, where do these pesky irrationals come from?
Tis a mystery of religious proportions.
I finish with the proof (by contradiction) that √2 is irrational. Enjoy!!!
Think of the number line. It seems reasonable, doesn't it dear reader, to assume that you can keep subdividing it up using fractions until you fill it completely.
Just take the gap between 0 and 1. Surely we can pepper it with fractions. 1/2, 1/4, 1/8, 1/16, 99/100, 999/1000 and so on and so forth.
And using fractions we can always put another point between two points ad-infinitum. So that this ............ eventually becomes ________ !!!
So, where do these pesky irrationals come from?
Tis a mystery of religious proportions.
I finish with the proof (by contradiction) that √2 is irrational. Enjoy!!!
4 comments:
A short proof of this result is to obtain it from rational root theorem, that if p(x) is a monic polynomial with integer coefficients, then any rational root of p(x) is necessarily an integer. Applying this to the polynomial p(x) = x2 − 2, it follows that √2 is either an integer or irrational. Since √2 is not an integer (2 is not a perfect square), √2 must therefore be irrational.
See quadratic irrational for a proof that the square root of any non-square natural number is irrational.
It is not known whether √2 is a normal number, a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that is normal to base two :)
your talking my language!
and aint wikipedia a blast?
it would be if I didn't know what I was talking about, but I want to believe that 3 years of civil engineering studies weren't in vain. In my defense I had to copy and paste because of my English language limitations :)...
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