It remains an unsolved mystery for mankind.

π is a very important number in mathematics. I believe everyone knows the meaning of pi (the ratio of the circumference of a circle to its diameter) , but do you know how to calculate the value of pi? Do you know that you can also calculate pi using a needle or millet that is commonly found at home?

1

Classical method of calculating pi

The ancients realized a long time ago that the ratio of a circle's circumference to its diameter is a constant, and they made a rough measurement of this value. The measurement method was to directly measure the circumference and diameter of the circle separately and then compare them.

However, because the circles drawn in ancient times were not perfect circles and the measurement accuracy was insufficient, the value of π obtained by this method has a large error. In the Tang Dynasty, Yang Jiong wrote in his article "Huntianfu": " The circumference of the three circles is one, and the diameter is different from the celestial pole; the east well and the south dipper are different from the Milky Way in terms of curvature and straightness ." It can be seen that ancient people believed that π=3.

In fact, as early as the Three Kingdoms period, Chinese mathematician Liu Hui invented a method to accurately calculate pi: the method of dividing a circle . This is also the first iterative algorithm in Chinese mathematical history to mathematically calculate pi to any degree of accuracy.

Principle of the circle division method: Green is a hexagon, blue is a dodecagon. It can be seen that the area of ​​the dodecagon is closer to the area of ​​the circle. If the number of sides continues to increase, its area will be closer to the circle.

Image source: wikipedia

Liu Hui's method of dividing a circle is based on the circle area calculation formula S=πR².

In the circle division method, Liu Hui applied the idea of ​​limit. He believed that if the circle is divided into polygons as shown in the figure above, the finer the division, the more sides the polygon has, and the area of ​​the polygon is getting closer and closer to the area of ​​the circle, until there is no difference in the end . Then, by calculating the area of ​​the polygon, we can get the value of π.

Zu Chongzhi, a famous mathematician during the Northern and Southern Dynasties, used Liu Hui's method of dividing a circle into 12,288 polygons 11 times, and obtained the value of pi, π=3.1415926, which was the most accurate value of pi in the world for nearly a thousand years.

2

These methods of calculating pi are quite interesting

In addition to using geometric methods, there are also some interesting ways to calculate pi, such as using a needle or millet to calculate pi as mentioned in the previous article.

In the 18th century, mathematician Buffon proposed the following problem: Suppose we have a floor with parallel and equidistant wood grain (as shown below). Now throw a needle at random whose length is smaller than the distance between the wood grains. Find the probability that the needle will intersect one of the wood grains. This is the Buffon needle problem .

Buffon's needle problem Image source: Wikipedia

The answer to Buffon's needle game requires some knowledge of probability theory and calculus, so this article will not describe the derivation process in detail. If the length of the needle is l, the length between the parallel lines is t and t>l, we can get the probability that the needle and the pattern intersect:

In the actual needle throwing process, if we throw the needle n times, and h of them only intersect with the grain, then at this time, we can know that the more needles are thrown, the more accurate the calculated π will be.

Since this method requires throwing needles many times to calculate the value of π, it may be dangerous. So, let me introduce a method to calculate π with zero risk that can be done with a piece of paper and millet - the Monte Carlo method using the circle area formula .

Suppose we have a square piece of paper with a side length of 1, and draw a quarter circle on the paper with one vertex of the square as the center and the side length of the square as the radius. Then if we randomly select a point on the square, what is the probability that this point is within the quarter circle?

I believe that smart readers have already given the answer to this question, which is the area of ​​the quarter circle is greater than the area of ​​the square, that is, P = π / 4. If we throw n points, and h of them are in the quarter circle, then we can know.

Randomly throwing points to estimate the value of π Image source: wikipedia-nicoguaro

However, in order to obtain a sufficiently accurate value of π, the number of tosses n needs to be very large, so this experiment is usually conducted on a computer. If we use millet and paper to conduct this experiment, it may take a long time to count the millet (of course, it is also a challenge to our eyesight).

3

Pi, everywhere

Pi has a very important meaning in mathematics, not just for calculating the area of ​​a circle. There are many times when pi suddenly appears in problems you don't expect. For example, a well-known problem in mathematics: the Basel problem .

The so-called Basel problem is to find the sum of the following series:

.

This problem was first proposed by Pietro Mengoli in 1644 and solved by the great mathematician Euler in 1735. People can easily calculate that the sum of this series is approximately equal to 1.644934.

Mathematicians had never thought that this series would have anything to do with π. However, Euler gave a proof in 1735 that the sum of this series is π. This surprised the mathematical world and Euler became famous.

This series was later generalized by Riemann, who defined the Riemann zeta function, which is the essence of the " Riemann hypothesis ", one of the biggest problems in mathematics.

4

Modern method of calculating pi

After reading the previous section, some readers may have thought of this: can we use this formula to calculate π?

After all, calculating the reciprocal of the square of a natural number seems to be easier than cutting a circle, and more reliable than throwing a needle or millet. The answer to this question is of course yes. Nowadays, the calculation of π is solved by applying the series method.

However, the effect of using series to calculate π was not very good. The accuracy of calculating hundreds of terms of π was not as high as Zu Chongzhi. At this time, the appearance of a genius changed this phenomenon. He was the mathematical genius: Srinivasa Ramanujan .

He is accustomed to deriving formulas based on intuition (or skipping steps or calling it number sense) and does not like to do proofs, but his theories are often proven to be correct afterwards (students and friends should not try to learn from him, as you will not get any points in the exam if you do).

Ramanujan made great contributions to mathematics, but unfortunately he died at the age of 32. His early death, like Galois's death at the age of 20 and Abel's death at the age of 26, is a great loss to the mathematics community.

Why is he considered a mathematical genius? Let’s look at a few formulas that he claimed to have “dreamed of”.

Some formulas given by Ramanujan

Based on Ramanujan's work, mathematicians proposed the commonly used formula for calculating pi: the Chudnovsky formula . Using this formula, calculating one term can give more than a dozen terms of π. Now mathematicians have used this formula to calculate 62.8 trillion digits after π.

Chudnovsky formula

In addition, there are some very interesting formulas for calculating pi, such as the Bailey-Bohlwin-Plouffe formula (BBP formula) , which can calculate any digit of pi in hexadecimal without calculating the previous digits, making collaborative calculation of pi possible.

BBP formula

5

Can pi be calculated completely?

From ancient times to the present, mathematicians have expected that π will have some special properties, such as being calculated to the end, repeating after a certain position, or being expressed as some simpler algebraic expressions.

However, this hope was severely shattered by the group theory founded by Galois mentioned in the previous article. This theory shows that π is a transcendental number , that is, π is not the root of any algebraic equation, and it cannot be expressed in the form of an algebraic expression composed of algebraic numbers of finite length. We can only use the infinite series or integrals mentioned above to accurately represent the value of π.

However, the mathematical community has a new conjecture about π. They believe that π is a " normal number ", which means that the probability of each number appearing in π is equal. This conjecture has not been proven.

However, computer scientists have proved through exhaustive enumeration that π contains all 8-digit numbers. This means that our birthdays, our graduation ceremonies, our wedding anniversaries... all dates will appear in π. Why not check now which number your birthday belongs to in π?

6

Do we need to know pi?

The current calculation of pi has actually far exceeded the scope of practical application. The error of calculating the circumference of a circle equal to the radius of Pluto's orbit using pi with dozens of digits is already less than the scale of an atomic nucleus.

Currently, the calculation of pi is mainly to test the computing power of supercomputers . Like the search for Mersenne primes and twin primes, the calculation of pi is a "big test" that supercomputers must go through. However, even the most powerful computers cannot completely calculate π. There are still infinite secrets hidden in π, waiting for humans to explore.

Perhaps one day in the future, mankind can proudly tell Liu Hui, Zu Chongzhi, Euler, Ramanujan and many other predecessors: "We have fully understood π."

This article is produced by Science Popularization China and supervised by China Science Popularization Expo

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