Root 2 – Numberphile

The square root of two is a fascinating number with a long and sordid history. It also forms the basis of most office paper, such as A4, A3, etc.
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This video features Professor Roger Bowley and Dr James Grime.

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How can the long side divided by the small side equal sqrt(2)? sqrt(2) is irrational which means no two integers a and b can divide such that a / b = sqrt(2). The A paper series has nothing to do with the sqrt(2)

I'm conducting an experiment in neural bio-chemistry & math. Math is a terrible subject for me but I'm going to watch a large swathe of these videos & see if it will improve my skill through immersion. Sixty Symbols & PTV improved my general scientific knowledge greatly.

Ehm so you can't express the square root of 2 in a fraction but at the start he said, when you divide the one side of any "a" paper by the other side of the paper then you get the square root of 2, wouldn't that be a fraction ?

With root 2 being irrational, I wonder to how many decimal places the length and width of an A4 sheet of paper are accurate? If we can't get to the last digit of root 2, is it correct that we also can't get to the last digit of what the precise measurements of both the length and width of the sheet would need to be in order for one divided by the other to give root 2?


from egypt , we have new Scientist he Explain It To Us In He's Program Its ( El daheeh ) Mr : Ahmed El ghandour i understand it from him ,Easily regardless of your explanation with all respect <3

The basic problem with this proof is that the formulation "let √2=a/b where a and b are both integers and a/b is in simplest terms" is a combination of three separate conditions. The first one, that √2=a/b, is the starting point for the proof so that's fine and the second one, that a and b are both integers, is used when asserting the evenness of a and b (because if you double something that isn't an integer the result might not be even), but the third condition, that a/b is in simplest terms (i.e. that a and b are co-prime) is as far as I can see NOT used in any step of the method until the final "contradiction" and so is in effect arbitrary. For example, suppose I said "let √4=a/b where a and b are both integers", but then imposed a third condition that a must be odd, I will get a contradiction because I can prove in a few lines that a is even.

So what is actually being proved here? If you drop the arbitrary third condition then it's NOT proving that √2 is irrational as such, but rather, because the doubling of a and b can carry on indefinitely, it's proving that √2 is the limit of some rational sequence involving powers of 2. So row me out in the boat with Hippasus if you must, but personally I'm quite happy that the limit of a rational sequence can be √2 as this was known even in antiquity (the so-called "Heron's method").

actually, can't you just take root(2) = a/b, 2 = a^2/b^2, a^2 = 2 and b=1 (simplest terms) and therefore root(2)'s simplest form is just root(2)/1 and therefore it can't be a fraction?

Does math still have the same religious or spiritual significance it used to have then for people like the Pythagoreans?
It is rather easy to poke fun and ridicule some of the beliefs of ancient times and cultures, however we will do well to notice that even today fact does not always have and easy path.
Also one may question as to whether we are becoming more open minded or more prone towards group think?

1:45 Argh… One square metre NOT one meter squared. The latter is an exact square with side 1m. The former is any area equal in size but can be any shape.

Yes, when talking about one it's not that big of a deal but as soon as you are not using one, it's important.

Two metres squared is 4 square metres.

Maybe they persecuted Pythagores because he was a vegan. I find that the meat and dairy industry has been a corrupt and evil organization since the enslavement of mankind

Ok the sqrt(2) is irrational , then how can a ratio equal root(2) ?
Since you cant write an irrational number as a two number devided to each other?

I work in the printing industry and it is pretty common to round off at certain decimals.
A0 = 1 188mm × 840mm
A1 = 840mm x 594mm
A2 = 594mm × 420mm
A3 = 420mm × 297mm
A4 = 297mm × 210mm
A5 = 210mm × 148mm
A6 = 148mm × 105mm
A7 = 105mm × 74mm
A8 = 74mm × 52mm
A9 = 52mm x 37mm

You'll notice 148 isn't half of 297
148,50 is.
Like wise the short side of an A7 should be 74,25mm

It may seem petty, but calculating further on the practice of rounding off these sizes we'd end up with A20 being a square.

So while in formula the A-sizes will always be √2, in practice not so much.
Though the inner mathematician of me keeps the accuracy to the 2nd decimal after the comma, it is impossible to keep paper in exact ratios when cutting them in half, because it's paper.

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