E-StoreIN PURSUIT OF PI
15 Aug 2008 09:53:17 IST
IN PURSUIT OF PI
In honor of Pi Day 2008, I thought I would post a little article about the remarkable, infinite, and irrational Pi.
So what is Pi, anyway? Simply put, it's the ratio of a circle's circumference to its diameter. So take a circle, say the lid of a jar. Measure the diameter (i.e., the width of the circle) and then measure the circumference. Divide the circumference by the diameter and you'll get an approximation for Pi.
This happens no matter the size of the circle. If a circle has a diameter of 1, then the measure of the circumference is Pi. When you make these measurements by hand, you'll end up approximating Pi as 3.1 or 3.2. But if measured precisely -- and I mean impossibly precise -- you get a decimal after the 3 that never ends and never repeats. That is to say, Pi is irrational. There are no two numbers we can write as a fraction to express Pi. For example, 3/2 = 1.5. Therefore 1.5 is a rational number. There are no corresponding pair of numbers we can use to express Pi. Some come close, such as...
But the true value of Pi is 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510....
The series of dots at the end mean the decimals continue into infinity.
It has taken mathematicians literally hundreds of years to approximate Pi this accurately. Archimedes was the first to make a serious run at calculating Pi. Using geometry around 200 BC he was able to determine that Pi was greater than 223/71 but less than 22/7. About 600 years later, the Chinese mathematician Zu Chongzhi would improve on this and show Pi to be greater than 3.1415926 but less than 3.1415927. This would stand for the next 900 years as the most accurate estimation for Pi!
In 1400, the Indian mathematician Madhava discovered the following formula for Pi, which was then later rediscovered by James Gregory and Gottfried Leibniz in the 17th century:
Check out those dots at the end! We are dealing with a series of infinitely many terms. Go out about 4000 terms and you start to improve on the estimations of Archimedes and Chongzhi.
In the 16th century, Francosi Viete discovered the following formula for Pi comprised entirely of the number 2:
Don't miss the dots on the end! The more calculations you carry out, the better your approximation of Pi. But be prepared to do a lot of calculations. This formula converges on Pi very slowly. And the square root of 2 is itself irrational, so you'll never get a precise answer. But it is a pretty formula to look at!
In the 17th century, John Wallis discovered another formula for Pi:
Again, notice the dots on the end. This is an infinite product. The more calculations you do, the better your approximation to Pi. But like Viete's formula, convergence on Pi is slow. Even with 60 terms in the calculation, the approximation of Pi is only good to one decimal place.
In the 18th century, yet another John, John Machin, discovered yet another way to approximate Pi:
Now this formula is actually worth playing with! Get out a pencil, some paper and your calculator and try a few terms. You can make some very good approximations of Pi. (By "good", I mean accurate to several decimal places).
Now the formulas above are very nice. They're true and they work. But for sheer power and accuracy, they can't touch the ultimate formula for Pi:
What you see above is the work of Srinivasa Ramanujan (1887-1920). How powerful is his formula? I'll show you. Notice the Greek sigma in the formula? That tells you to do the following....
So you let k = 0 and calculate the value.
Then you let k = 1 and calculate the value.
Then you let k = 2 and calculate the value. etc....
And then you add them all together.
BUT...throw out the sigma. Let's make it "simple" and just do it ONE TIME with k = 0. We get the following approximation:
The correct answer is...
With only ONE iteration, the approximation is accurate to SEVEN decimal places! This is truly a powerful formula!
If you like math and Pi, you owe it to yourself to learn more about Srinivasa Ramanujan. With no formal academic training and isolated in colonial India, Ramanujan made astounding mathematical discoveries. We are very fortunate to know of his existence at all. Fortunately he sent some of his work to a British mathematician, G.H. Hardy, who recognized his prodigious talents and brought him to England for formal study and academic work. But tragically, Ramanujan died only 5 years later at the heart-breaking young age of 32.
I hope this article has given you a little deeper appreciation of Pi. It's a lovely number that has pushed some of mankind's greatest minds to remarkable achievements. Memorize a few digits and take a moment to reflect.
Happy Pi Day 2008!
Clay
This happens no matter the size of the circle. If a circle has a diameter of 1, then the measure of the circumference is Pi. When you make these measurements by hand, you'll end up approximating Pi as 3.1 or 3.2. But if measured precisely -- and I mean impossibly precise -- you get a decimal after the 3 that never ends and never repeats. That is to say, Pi is irrational. There are no two numbers we can write as a fraction to express Pi. For example, 3/2 = 1.5. Therefore 1.5 is a rational number. There are no corresponding pair of numbers we can use to express Pi. Some come close, such as...
But the true value of Pi is 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510....
The series of dots at the end mean the decimals continue into infinity.
It has taken mathematicians literally hundreds of years to approximate Pi this accurately. Archimedes was the first to make a serious run at calculating Pi. Using geometry around 200 BC he was able to determine that Pi was greater than 223/71 but less than 22/7. About 600 years later, the Chinese mathematician Zu Chongzhi would improve on this and show Pi to be greater than 3.1415926 but less than 3.1415927. This would stand for the next 900 years as the most accurate estimation for Pi!
In 1400, the Indian mathematician Madhava discovered the following formula for Pi, which was then later rediscovered by James Gregory and Gottfried Leibniz in the 17th century:
Check out those dots at the end! We are dealing with a series of infinitely many terms. Go out about 4000 terms and you start to improve on the estimations of Archimedes and Chongzhi.
In the 16th century, Francosi Viete discovered the following formula for Pi comprised entirely of the number 2:
Don't miss the dots on the end! The more calculations you carry out, the better your approximation of Pi. But be prepared to do a lot of calculations. This formula converges on Pi very slowly. And the square root of 2 is itself irrational, so you'll never get a precise answer. But it is a pretty formula to look at!
In the 17th century, John Wallis discovered another formula for Pi:
Again, notice the dots on the end. This is an infinite product. The more calculations you do, the better your approximation to Pi. But like Viete's formula, convergence on Pi is slow. Even with 60 terms in the calculation, the approximation of Pi is only good to one decimal place.
In the 18th century, yet another John, John Machin, discovered yet another way to approximate Pi:
Now this formula is actually worth playing with! Get out a pencil, some paper and your calculator and try a few terms. You can make some very good approximations of Pi. (By "good", I mean accurate to several decimal places).
Now the formulas above are very nice. They're true and they work. But for sheer power and accuracy, they can't touch the ultimate formula for Pi:
What you see above is the work of Srinivasa Ramanujan (1887-1920). How powerful is his formula? I'll show you. Notice the Greek sigma in the formula? That tells you to do the following....
So you let k = 0 and calculate the value.
Then you let k = 1 and calculate the value.
Then you let k = 2 and calculate the value. etc....
And then you add them all together.
BUT...throw out the sigma. Let's make it "simple" and just do it ONE TIME with k = 0. We get the following approximation:
The correct answer is...
With only ONE iteration, the approximation is accurate to SEVEN decimal places! This is truly a powerful formula!
If you like math and Pi, you owe it to yourself to learn more about Srinivasa Ramanujan. With no formal academic training and isolated in colonial India, Ramanujan made astounding mathematical discoveries. We are very fortunate to know of his existence at all. Fortunately he sent some of his work to a British mathematician, G.H. Hardy, who recognized his prodigious talents and brought him to England for formal study and academic work. But tragically, Ramanujan died only 5 years later at the heart-breaking young age of 32.
I hope this article has given you a little deeper appreciation of Pi. It's a lovely number that has pushed some of mankind's greatest minds to remarkable achievements. Memorize a few digits and take a moment to reflect.
Happy Pi Day 2008!
Clay
Free Sign Up!
Preparing for IIT-JEE ?
Arihant Revision Package for IIT JEE - Books, Practice Tests + Rank Predictor
@ INR 1,995/-
For Quick Info
|
|
|
|
|
| 35,229,755 Engineering Questions Attempted |
6,69,530 Visitors Monthly |
79 Engg. Exam Covered |
2361 Students Currently Online |
Connect with Us  |
 |
Vriti Communities
Privacy Policy © Vriti 2011