The mysterious π has been calculated to 100 trillion digits. How many secrets are there behind it?

Pi (abbreviated as π) is one of the most mysterious and familiar numbers in mathematics. Whether in architecture, engineering, aerospace, or physics, π plays a vital role. Simply put, pi is the ratio of the circumference of a circle to its diameter. It seems easy to get this number. Just measure the circumference of a circle and divide it by the diameter to get pi.

But the difficulty is, what kind of ruler can be used to measure the circumference of this circle very accurately? From ancient times to the present, thousands of years have passed, and this vexing problem has not been completely solved. The result is that pi is an irrational number, which means that its decimal part is infinite, never-ending, and has no cyclic pattern.

So how many decimal places does this have? No one knows, because so far people only know that it is infinite, and infinity is endless. Thousands of years ago, ancient people tried to measure the exact circumference of a circle by hand. They used rulers and geometric methods to manually calculate π and got a rough data, which was finally accurate to nine decimal places, that is, 3.141615926. This is what our ancestor Zu Chongzhi got, and we can be proud of it.

Today, with the advancement of science, supercomputers have calculated π to 100 trillion digits, but this can only mean that it is still an approximate value and has not been exhausted. What is the concept of 100 trillion digits? If a person starts counting from birth, does not eat or drink, and counts two numbers per second on average, it will take more than 1.58 million years. In other words, a person has been counting since the primitive ape ancestors, and has continued to count for generations, and has not yet finished.

So, does pi really have so many digits? How did humans get it in the beginning? Let's find out.

Ancient wisdom: Calculating π using the circle-slicing method

Whether in the West or the East, as early as ancient times, there was a group of wise men who became interested in pi and began to study it. The East and the West combined different paths to get the approximate value of pi. Specifically, there are some typical representatives:

Archimedes' "polygonal approximation"

One of the first mathematicians to systematically calculate π was Archimedes of ancient Greece (c. 287-212 BC). His idea was simple:

1. Draw a circle, then draw a regular hexagon inside the circle, and then draw another regular hexagon outside the circle.
2. By calculating the perimeters of these two hexagons, we can get an upper and lower limit for π.
3. Then, double the number of sides of the hexagon to make it a dodecagon and calculate the perimeter again.
4. Continue doubling it to become a 24-gon, a 48-gon... until it approaches the true circumference of the circle.

Using this method, Archimedes estimated the value of π to be between 3.1408 and 3.1429, which was an amazing achievement in an era without computing tools!

Zu Chongzhi: Calculations with an accuracy that is 1,000 years ahead of the world

Zu Chongzhi was an ancient Chinese mathematician during the Northern and Southern Dynasties (5th century AD). He improved the method of dividing a circle by splitting a positive circle into 24,576 sides. By measuring and calculating these 24,576 polygons, he obtained an approximate value of π with higher accuracy than Archimedes: 3.1415926 to 3.1415927. This accuracy was 1,000 years ahead of the world!

But the problem is that the circle-slicing method is very slow in calculating π. If you want a more accurate π, you have to draw polygons with more sides, and the amount of calculation will increase exponentially, and eventually become unbearable. Therefore, in the 1,000 years after Zu Chongzhi, the accuracy of π could no longer be improved. In modern times, mathematicians began to look for more efficient calculation methods to greatly increase the calculation speed of π.

The power of mathematical formulas: π can be accurately calculated without drawing pictures, and it is much faster than the method of dividing the circle

Ancient people calculated π by approximating the circle with its shape, while modern mathematicians use formulas to directly calculate π. This method is much faster and more accurate than the method of cutting the circle. The more representative formulas are:

18th century: Machin's Formula

In 1706, British mathematician John Machin proposed a formula for quickly calculating π:

π=16tan⁡−1(1/5)−4tan⁡−1(1/239)\pi = 16 \tan^{-1} (1/5) - 4 \tan^{-1} (1/239)π=16tan−1(1/5)−4tan−1(1/239)

This formula allows mathematicians to quickly calculate π using series expansion, without having to use the ancient method of cutting the circle. From the 18th to the 19th century, mathematicians continued to improve this type of formula, such as:

• John W. Wrench and others manually calculated π to 808 digits (1946), which was a world record at the time.
• Manually calculating the 100-digit value of π would take several months, with a high error rate. Gauss-Legendre Algorithm - Doubled Precision
• First, choose two numbers, one representing the radius of the circle and the other representing the perimeter of the polygon.
• These two numbers are constantly adjusted using mathematical formulas to make them closer and closer to the true value of π.
• Each time this method is used, the accuracy of π can be doubled, and the calculation speed is much faster than the squaring of the circle. But before the advent of modern computers, even with better formulas, the accuracy and speed of π calculations were much faster than the squaring of the circle, and traditional manual calculations were still very complicated and slow. It was not until 1948 that Ferguson of the UK and Wrench of the US jointly published the 808- digit decimal value of π, which became the highest record of manually calculated pi. The advent of modern computers has brought about a qualitative leap in the calculation of π.

By the 1970s, mathematicians had discovered a way to increase precision exponentially:

• First, choose two numbers, one representing the radius of the circle and the other representing the perimeter of the polygon.
• These two numbers are constantly adjusted using mathematical formulas to make them closer and closer to the true value of π.
• This method can double the accuracy of π each time it is calculated, and the calculation speed is much faster than the method of dividing the circle.

But before the advent of modern computers, even with better formulas, the accuracy and speed of calculating π was much faster than the method of squaring the circle, but traditional manual calculations were still very complicated and slow. It was not until 1948 that Ferguson of the UK and Wrench of the US jointly published the 808- digit decimal value of π, which became the highest record of manually calculated pi. The advent of modern computers has brought about a qualitative leap in the calculation of π.

**Computer calculation of π: ****In 1949, the first computer calculation of π was carried out, **and the efficiency improved by leaps and bounds

In 1949, Americans John von Neumann and Nicholas Metropolis used an electronic computer to calculate π for the first time. The computer they used was ENIAC (Electronic Numerical Integrator and Computer), and the calculation reached 2037 digits, breaking all manual calculation records in human history.

Computational efficiency improvement comparison:

• It would have taken months to calculate 808 digits of π by hand, but ENIAC calculated 2037 digits in 70 hours, increasing the speed by hundreds of times.
• ENIAC could perform 5,000 additions per second, far exceeding manual calculations. This calculation marked the first breakthrough in mathematical calculations by computers, and also brought the calculation of π into the modern computing era.

1960s-1980s: Computer Pi calculations exceed one million digits

With the development of computer technology, mathematicians began to use more efficient algorithms, such as the Gauss-Legendre Algorithm, which made the calculation speed of π increase exponentially. The calculation efficiency increased from 70 hours for ENIAC to calculate 2037 bits in 1949 to only 5 hours for CRAY-1 to calculate 8 million bits in 1982, which increased the speed by more than 100,000 times.

1990s to present: Supercomputers calculate π

In 1987, mathematicians the Chudnovsky brothers proposed a superfast calculation method:

1π=12∑k=0∞(−1)k(6k)!(545140134k+13591409)(3k)!(k!)3(640320)3k+3/2\frac{1}{\pi} = 12 \sum_{k=0}^{\infty} \frac{(-1)^k (6k)! (545140134k + 13591409)}{(3k)! (k!)^3 (640320)^{3k+3/2}}π1=12k=0∑∞(3k)!(k!)3(640320)3k+3/2(−1)k(6k)!(545140134k+13591409)

Although this formula looks complicated, its advantage is that it calculates extremely quickly!

• With each calculation, the precision of π can increase by 1 million digits!

• This is the most common method for calculating π by computer. After entering the 21st century, mathematicians began to use more efficient algorithms (such as the Chudnovsky algorithm) and supercomputers, and the calculation speed and accuracy of π were further improved. Manual calculation of 1000 digits of π takes months, but a computer can complete it in seconds. With the optimization of mathematical formulas and the continuous upgrading of computers, the calculation speed is getting faster and faster. The comparison of the improvement of calculation efficiency is as follows:

• In 1949, it took ENIAC 70 hours to calculate 2037 digits of π, while in 2022, a supercomputer will only take 157 days to calculate 100 trillion digits.

• The number of digits required for calculation has increased from 2,000 to 100 trillion, a 5 billion-fold increase, while the calculation time has only increased by about 1,000 times, indicating that the computing power of computers has been greatly improved. Computers have completely replaced manual calculations for the modern value of π. As of 2023, a Swiss research team has used a supercomputer to calculate π to 100 trillion (10¹⁴) digits, breaking the historical record. Today, the main bottleneck for our calculation of π is no longer the mathematical formula, but the speed and storage capacity of computer hardware. In the future, with the development of quantum computing, the calculation of π may become more efficient.

What is the point of calculating so many digits of π?

In fact, in daily life, the π we use usually does not exceed 3.1416. Even when NASA calculates the orbit of a spacecraft, it only uses 15 digits (3.14159265358979). However, calculating ultra-high precision π still has many meanings:

• Testing the performance of supercomputers: Calculating π requires a lot of computing resources and can test the processing power and stability of the computer.
• Mathematical Research: Mathematicians have wondered whether the infinitesimal part of π is truly completely random, or whether there is a hidden mathematical pattern.
• Science and Engineering: In some precise scientific research (such as quantum computing and black hole simulation), ultra-high precision of π is required. Although we may never need all 100 trillion digits of π, the process of calculating it has promoted the development of mathematics, computer science and engineering.

The calculation of the exact value of π is still ongoing, and there is no end in sight

Because π is an irrational number, its decimal points are infinite and non-repeating, which means that the search for the exact value of π is endless. From Archimedes's method of dividing a circle to the Chudnovsky algorithm of supercomputers, mankind's pursuit of π has lasted for more than 2,000 years. Today, the calculation accuracy of π has far exceeded actual needs, but scientists are still constantly challenging the limits, not only to calculate π, but also to promote the development of mathematics and computer science.

Therefore, π is not just a mathematical constant, it is a symbol of human wisdom and technological progress. In the future, humans will use more advanced methods to calculate π further, and perhaps uncover its deeper secrets!

This is an original article from Space-Time Communication. Please respect the author’s copyright. Thank you for reading.