How To Calculate Pi
What does the sigma notation mean?
Hi, I am Dr. Shah. I was the National Lecturer Competition winner in 1989, and I am the math master at Mathscool. Now, ready for a new variety of maths. So how do we calculate Pi? Well,if you press the PI button on your computer or calculate, you will find PI as 3.1415926535.and so on. It carries on and on and on. And it goes on and on forever, so you use it to whatever level of accuracy you require for your calculation. Sometimes 3.14 is good enough, and sometimes you need to be more accurate and you use 3.1416, so an engineer might be using 3.1416. Well, so how did mathematicians calculate this value of PI, and what are the ways in which you could calculate PI if you want to account more decimals places for it? Well, there are few different ways of working it out and I am just going to go through two simple ways. One way of working out is this, and this is called the harmonic series. This is complicated because of the sigma notation, but let's just break down what that actually means. This Sigma in the notation just simply means Plus. That's all it means, so what we are going to do with it is first of all, put R as 1 in the equation, so one over one square and this SIGMA means plus. And now we put the next number, 2 into that equation, 1 over 2 square and then Plus for the SIGMA and then 1 over 3 square and 1 over 4 square and so on, and you go on well as far as you want to go. It's up to infinity so the further you got, the more accurate it is. And that's equal to PI square over 6, so if you type 1 over one square, 1 over 2 square ,1 over 3 square and so on, then multiply the answer by 6, and then square root it, and then you will have PI. So that's one way of calculating PI. What about an alternative way for calculating PI? You might want to use this alternative formula and again, we are using SIGMA notation slightly more complicated to understand but it's not particularly difficult. So again, I'll just break it down for you. We have on this side a PI over 4 equals, so first, I am going to put R equal to 0 into this, -1 to the power 0 would be what? Anything to the power of 0 would be 1 over 2; lots of 0 plus 1 would be 1. So when you put 0 into that, it's just 1 over 1, so just 1. Now, I put in the next number -1 to the power 1. -1 over two loads of 1 plus 1 is 3, and this SIGMA means add, so I am going to put a PLUS in between them and carry on with the next one PLUS. So I put in 0, I put in 1, and so the next number I put in is 2. So -1 squared. When you square -1, it becomes 1 over 2, lots of 2 plus1 which is 5. By now, you can probably see the pattern. It's going to be 1 over 1, minus 1 over 3 plus 1 over 5 minus 1 over 7 plus 1 over 9 and so on and so forth, and that's equal to PI over 4. So work that out and then multiply the answer by 4, and that again would give you PI.