## Creative Café Vol 7 Grasp it with an approximate number 2/3pict

2010.09.29

**Nohara**：just before, you said that fractions were pretty, but expressing as decimals was a little ugly.

**Murakami**：yes, if you write down fractions which don't cycle, it is pretty; it's the truth-it really is pretty.

**Nohara**： I'm very interested

**Murakami**：this is pretty and this is pretty don't you think? (laughter). If you do this calculation with a calculator you can just repeat 'add 2' can't you? Just repeating putting in a figure, what's beautiful about that? One of the standards of beauty in our world is symmetry. If lines and points are symmetric then there is some element of beauty. In other words, repeating itself. A repeating pattern is somehow beautiful. The left side of this clearly repeats itself so is beautiful. Furthermore, when we ask what is good, this 2 gradually get smaller and disappears into the infinite distance. In this '......' is emotion, infinite emotion. (Laughter)

**Tsuda**：Mr Murakami, is this how we come to feel the beauty of Space? Because there is order?

**Murakami**：yes indeed. That is what is visible in what I circulated a little time earlier. Of course, the decimals I circulated also gradually get smaller and fade away. They fade away but if you look upon it from the point of view of the whole, the fact that the same things are repeating is a bit scary. When we think of something beautiful, there is often a scary aspect to the reverse side. Don't you think so? So, if you ask why 0.41421356... is hardly beautiful, it is because the way the numbers come out here is random and irresponsible. In reality they are not random but they appear so.

**Nohara**：you don't think that random conveys any feeling of beauty?

**Murakami**：if completely random, that might be the case but this is half finished randomness. If we talk about the square root of 2, it is the length of the diagonal of a square. With this meaning, even to infinity it is just this meaning and is hardly a beautiful shape. That's probably the mathematician; I'd like to change things which have no order at first glance in this way. Well, if I say again, when I said "one side is the diagonal of the square", even in this I felt a sense of ultimate beauty.

**N**：like a Russian doll

**Murakami**：Ah, yes- Matoroshika. But that's not infinite is it? Also, having looked at it what I don't like is that gradually the craftsmanship becomes very coarse. As for that, if the inside really was a reduced outside, I'd buy it whatever it cost. Absolutely, for instance here, there is a mistake. So, with the continuous fraction of the square root of 2, even if I start a Russian doll from wherever, it is completely the same. That is its infinite beauty and its infinite strangeness. Because there isn't any limit.

Obtaining Pi approximately

**Murakami**：okay let's move on. The next stage is similar. This time let's try adding and subtracting the reciprocal of odd numbers. Therefore starting with one, the next is to subtract 1/3. The next odd number is five so we add 1/5. The next is to subtract 1/7, then to add 1/9 and so on. If we carry on doing this over and over again, where do we end up? It's not so easy to answer. If we go from 1 to 201, adding and subtracting 100 odd numbers we get 0.78787878335. But if we repeat the process a thousand times, in other words from 1 to 1/2001, the answer is 0.7856....... If you do the calculation over and over again, it should become this number, but what actually is it?

**Q**：Since this is a science cafe, please make it a little more easy to understand.

**Murakami**：So what happens if I multiply this number by four. 0.785x4=3.14. It becomes a very familiar number. Really, if you keep on adding and subtracting for ever, and then multiply by four you get Pi. Adding and subtracting odd numbers. That's easy to remember isn't it (subtract 1/3, add 1/5, subtract 1/7, add 1/9 and so on)? If you carry on doing for a lifetime you will get Pi. The other day, competing with the speed of a supercomputer, I carried out column after column of calculations, and you don't really need to rely on a supercomputer, you can do it yourself. Go-ahead give it a try. Waste your whole life on it! (Laughter) that's not a good idea! If you do it 100,000 times, still, the figures three or more below the decimal point are still changing. So don't carry on too long! For instance now it is barely 3.14. I imagine people who use the computer think that this should be accurate, but in reality the accuracy is really bad. It's not something that I would recommend, but it is a method that even they primary school pupil can use to calculate Pi. People who are able to count in the old Japanese way were able to calculate Pi much more cleverly.

This number we call Pi, compared with the square root of 2 we introduced earlier, is a much stranger number. The square root of 2, as I showed you before, if you add the reciprocal numbers of 2 (1/2 etc.), it can be written down much more prettily; there is a similar law for Pi. However although there is a similar way of calculating, compared with the square root of 2, you don't get the feeling of a cycle. There is no formula for working out the value of Pi. The fact that the formula (4/1-4/3+4/5....) equals Pi is not an explanation. For these reasons it is called a transcendental number. Perhaps even a transcendental irrational number. In other words, not just impossible but super impossible; a transcendental and irrational number. If we follow this through in this way it becomes very interesting. Well, let me see, with Pi, one aspect of its amusing side is why it emerges from the adding and subtracting.

**R**：does the supercomputer calculation of Pi follow the same principle?

**Murakami**：no, it's probably different. With such a slow convergence, I don't think you would do it that way. With a supercomputer, at the same time as raising the computer's performance, you have to improve the algorithm. Wouldn't you use computers for much more perplexing problems? I don't know but if you look it up in Wikipedia, maybe the trade secret will come out. If you're looking to calculate Pi, rather than do it this way, it's better to use a piece of string around a circular object and measure it in order to get a pretty accurate figure. (Laughter) rather than these horribly complicated calculations, a piece of string is much quicker!