Squaring the Circle - Numberphile

Key Takeaways

right so today we're going to talk about one of the great unsolved problems in mathematics that went back to the ancient Greeks thousands of years ago which was eventually solved in 1882 and it's the problem is called squaring the circle and you may have even heard of the problem as a metaphor for something that's impossible to do the question is can you make a square with the same area as a circle now you have to understand the rules in ancient Greece they didn't have algebra so they could only construct numbers using lines and circles so you could only make things using a straight edge like a ruler but not a measured ruler just a straight edge and a compass so lines and circles those are the rules those are the rules that the ancient Greeks had to work by using those rules can you construct a square with the same area as a circle let's have a look at what you can do with rules and Compasses you can add numbers here's a line and it has length a then I add another line of length B and the whole thing is A + B so you can add numbers together quite easily with lines you can subtract numbers as well if I start with b and then I Mark half a length which I'm going to call a then this bit here that's going to be B minus a uh we can multiply if I draw a little triangle here it's do like that if this has length one and this has length a I'm going to scale up the triangle like this if I scale it up so now that this has length B the big triangle has length B then this has been scaled up as well so that this has a length a * B you can divide same idea but it's kind of the reverse of that I took a bigger triangle if that had a length of B and the long edge here has a length of a and if I scale it down so now this has a length of one then this has been scaled down as well and you scale it down so it has a length of a / B so you can divide by scaling triangles as well and there's one more thing you can do if I draw a line here this is a I'm going to add one to it so that's one and then I'm going to draw a semicircle and this length here is the square root of a and that's all you can do that is it with a with a ruler and a compass you can add multiply Divide Subtract work out square roots and that is it that is all you can do the numbers you make using those rules are called constructible numbers so all those numbers are just repeated use of those few rules if we had a circle if we had a circle let's make this easy let's say this circle has a radius of one the area of the circle is pi times the radius squared and the radius squared well that's one so the area of the circle is just Pi so I want a square with the same area I want a square now that has an area of Pi and you may be ahead of me here because if a square has an area of Pi the lengths of the sides are the square root of Pi now it's the square root of Pi now we can do square roots that's allowed so there's nothing yet to say that we can't do this so what's the problem well we know that Pi is irrational actually that wasn't easy to work out we only worked that out quite recently by quite recently I mean 1761 but being irrational is not enough there are constructible numbers that are irrational uh you can make the square root of two quite easily uh and you can make the square root of the square root of two and and so on so being irrational is not enough to say that this can't be done if I'm allowed to make this a bit more General if I'm allowed to use other sort of uh Roots like cube roots or fifth roots or you nth Roots if I'm allowed to add those in as well then numbers that you can make using those rules are called algebraic numbers that's just a bit more General same idea and it turned out that they were able to prove that Pi was not an algebraic number and if it's not an algebraic number then it's definitely not a constructible number now this is what I like the word for a number that is not algebraic is called a transcendental number which is a fantastic word now to prove something is Transcendental it's really hard uh in fact mathematicians had shown that transcendental numbers existed seven years before they had even found an example of a transcendental number so you have to imagine in those seven years people were saying to the mathematicians all right so show me a transcendental number come on show me a transcendental number but it was a really hard thing to do eventually they were found they found a few examples of this but we still had to wait until we were able to prove that Pi itself was transcendental and then that was done in 1882 once we've shown that we've shown that Pi is not constructable and that is a proof that you cannot Square the circle can you only not Square the circle given the parameters of these rules in this game could a computer square a circle yeah so you can you can make a square with sides that have length root Pi so absolutely so in terms so to the modernday people with with our tools of algebra right algebra is fantastic so people complain about algebra and having to learn algebra at school but if we didn't have algebra we would be doing maths like the ancient Greeks did maths I promise you that would be much much worse your textbooks would be ridiculous right and they' be very wordy uh algebra is an amazing powerful tool in mathematics it's not there to cause you problems it's there for a reason it's brilliant so we it's this is easy to make a square with uh the area of Pi but using the Greek rules straight lines and circles it's impossible to do you say it's easy to make a square with the area Pi but because Pi has this nature where the digits sort of continue and we can't even get to the exact figure surely you can't make a square that's exactly Pi because you don't even know what pi is or you can do it can you you can do it and you're you're treading on something called know's Paradox there which we're about to do next okay so that's so for a second video in a row we're going to tell people they have to wait to find out more if that's the way you want to play it Brady it is your choice this is Brady's decision right he's stringing you along here the representatives actually passed it unanimously 67 to0 they passed it now this is the weird bit on the same day that they were passing the bill