Squaring the Circle - Numberphile

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I would say that since ancient mathematics did not allow for infinite sums, they could not accurately construct pi. Even though sqrt(2) can be constructed, it can also be given an infinite sum. Pi is no different. Thus with some added rules, pi can easily be constructed.

I am not the biggest fan of Numberphile. When he says that the ancients didn't have algebra, he is only technically correct. They didn't have symbolic algebra, but they for certain manipulated unknown values using geometry. Moreover, the introduction of abstraction in algebra does not guarantee that we can find pi. As I said above, we need to use more complicated tools, like sums, that need more complicated topics, like infinity.

👍︎︎ 2 👤︎︎ u/496ycm 📅︎︎ Mar 16 2014 🗫︎ replies

At minute 5:33

👍︎︎ 1 👤︎︎ u/groggyrat 📅︎︎ Mar 15 2014 🗫︎ replies

Can you clarify what you mean?

👍︎︎ 1 👤︎︎ u/Orophin 📅︎︎ Mar 15 2014 🗫︎ replies
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DR. .JAMES GRIME: 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 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 the 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 the Ancient Greeks had to work by. Using those rules, can you construct a square with the same area as the 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. And I add another line of length b. And the whole thing is a plus 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 off a length, which I'm going to call a, then this bit here, that's going to be b minus a. We can multiply. If I draw a little triangle here, this has length 1. And this has length a. I'm going to scale up the triangle. I scale it up, so now 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 times b. You divide. Same idea, but it's the reverse of that. If I took a bigger triangle, if that has a length of b, and the long edge here has a length of a, if I scale it down so now this has a length of 1, then this has been scaled down as well. And you scale it down so it has a length of a divided by 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 1 to it. That's 1. I'm then 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 ruler and a compass, you can add, multiply, divide, subtract, workout square roots, and that is it. That is all you can do. The numbers you make using those rules are called constructible all numbers. So all those numbers are just repeated use of those few rules. If we had a circle-- let's this make this easy. Let's say this circle has a radius of 1. The area of the circle is pi times the radius squared. And the radius squared, that's 1. 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 a square has an area of pi, the lengths of the sides are the square root of pi Now it's at the square root of pi. 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. You can make the square root of 2 quite easily. And you can make the square root of the square root of 2, 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 roots, like cube roots, or fifth roots, or nth roots, if I'm allowed to add those in as well, the numbers that you can make using those rules are called algebraic numbers. That's just a bit more general, same idea. And it turns 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 is really hard. In fact, mathematicians had shown that transcendental numbers existed seven years before they even found an example of a transcendental number. So you have to imagine in those seven years, people were saying to the mathematicians, show me a transcendental number, come on, show me a transcendental number. But it was a really hard thing to do. Eventually they found a few examples of this. But we still have 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 constructible. And that is a proof that you cannot square the circle. BRADY: Can you only not square this circle given the parameters of these rules in this game? Could a computer square a circle? DR. .JAMES GRIME: You can make a square with sides that have length root pi. Absolutely. So to the modern day people, with our tools of algebra-- algebra is fantastic. 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. They'd be very wordy. 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 this is easy to make a square with the area of pi. But using the Greek rules, straight lines and circles, it's impossible to do. BRADY: You say it's easy to make a square with the area of pi. But because pi has this nature where the digits 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? DR. .JAMES GRIME: You can do it and you're treading on something called Zeno's Paradox there, which we're about to do next. BRADY: OK. So for a second video in a row we're going to tell people they have to wait to find out more? DR. .JAMES GRIME: If that's the way you want to play it, Brady, that's your choice. This is Brady's decision. He's stringing you along here. The representatives actually passed it unanimously, 67 to 0. They passed it. Now this is the weird bit. On the same day that they were passing the bill--
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Channel: Numberphile
Views: 2,206,765
Rating: 4.9497285 out of 5
Keywords: squaring circle, square, circle, pi, numberphile
Id: CMP9a2J4Bqw
Channel Id: undefined
Length: 7min 34sec (454 seconds)
Published: Mon Mar 25 2013
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