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Review
In the world of mathematics, there is a seemingly mundane but mysterious number - pi. It is a treasure in the field of mathematics and hides the profound mysteries of mathematics.
You may have wondered why scientists keep calculating pi to more decimal places? This seems to be a mathematical "useless move", but in fact, there is more exploration and meaning behind it.
Magical pi
Pi, an important constant in this field of mathematics, is an infinitely non-cyclic decimal used to express the ratio of the circumference and diameter of a circle.
This magic number plays a key role in fields such as geometry, trigonometry, calculus, astronomy, and cryptography. So, how do we discover and calculate pi?
Dating back to the 20th century BC, we find that the ancient Babylonians recorded an approximation on a stone plaque that believed that pi was about 3.125.
At the same time, the ancient Egyptian Lyinder mathematical papyrus also reveals their study of pi, giving an approximation of 16/9 squared, which is about 3.1605.
The people of these ancient civilizations probably obtained these approximations by actually measuring the circumference and diameter of a circle, albeit with less precision, but have demonstrated their rudimentary understanding of pi.
In the 3rd century BC, the ancient Greek mathematician Archimedes used the first theoretical method to approximate the value of pi. He used an ingenious idea of using regular polygons to approximate the ratio of the circumference to the diameter of a circle.
He first attached a regular hexagon to a circle, and then another regular hexagon to a circle, and calculated the ratio of the circumference to the diameter of the two polygons, thus obtaining the upper and lower limits of pi.
Subsequently, he gradually increased the number of sides of the polygon, from hexagons to dodecagons, to twenty-four, and all the way to ninety-six, finally obtaining an approximation of pi between 3.1408 and 3.1429.
This method opened up a precedent for obtaining an approximation of pi through theoretical calculations, and provided inspiration for later mathematicians.
Then, in the 3rd century AD, the Chinese mathematician Liu Hui used a similar method to calculate the approximation of pi. He proposed a calculation method called circumcision, which was very similar to Archimedes' idea, but used more rounds, so that more accurate values were obtained.
He calculates the ratio of the circumference to the diameter of the two polygons by inscribed within the circle and by the outer circumference of the other regular ninety-six polygons.
In addition, he uses an ingenious algorithm that does not overcomplicate the calculations by increasing the number of edges. He ended up with an approximation of pi of about 3.14159, which is more accurate than Archimedes' results.
In the 5th century, another Chinese mathematician, Zu Chongzhi, further refined Liu Hui's method by calculating the approximation of pi to seven decimal places.
He calculated the ratio of the circumference to the diameter of the two polygons using a regular 12,288 polygon inscribed to a circle and another positive 12,288 polygon circumscribed to a circle.
He also used a more efficient algorithm to quickly find the side length of a regular polygon, which speeded up the calculation. Zu Chongzhi finally got an approximation of pi between 3.1415926 and 3.1415927, which was an astonishing achievement and set a thousand-year record for the calculation of pi.
How accurate is pi?
Since the 17th century, mathematicians have discovered many formulas for expressing pi, such as infinite series, infinite consecutive fractions, infinite products, etc.
The advantage of these formulas is that if you calculate enough terms, you can get an arbitrary-precision approximation of pi. Among them, the most famous Leibniz formula was independently discovered by the Indian mathematician Mahavira and the European mathematician Leibniz. It is in the form of: π=4(1-1/3+1/5-1/7+1/9-...... ）
Although this formula is simple and elegant, it converges slowly, which means that in order to obtain a high degree of pi, a large number of terms need to be calculated.
For example, to find pi to 10 decimal places, you need to calculate about 500 billion terms. As a result, mathematicians looked for formulas that converged faster, such as Chapman's, Forrest Gump, and Mahrin's formulas. These formulas are more efficient to calculate, for example, to get a value of 10 decimal places, you only need to calculate dozens of terms.
These formulas enable one to calculate more digits of pi using a computer. With the advancement of computer technology, the accuracy of pi calculation has also been greatly improved. Since the 40s of the 20th century, people began to use electronic computers to calculate pi, and initially only a few hundred digits of the results were obtained.
By the 21st century, that number has reached hundreds of billions. As of August 17, 2021, researchers in Switzerland have successfully calculated pi to 62.8 trillion decimal places using a supercomputer after 108 days of calculations. This is currently the highest record for human calculation of pi, and it is also a huge test of supercomputer performance and hardware.
But what's the point of calculating so many digits of pi? For practical applications in science and engineering, only a few tens or even fewer of pi are usually sufficient.
For example, to calculate the diameter of the entire observable universe, only 40 decimal places of pi are sufficient. So why do scientists keep calculating pi with more digits? This is because calculating pi is not only a challenge and exploration, but also a contribution to human civilization.
By calculating pi, we can explore the mysteries of mathematics, such as whether pi is a normal number, that is, whether its numbers after the decimal point are randomly distributed without any rules. In addition, practical applications can be found in areas such as statistical engineering, astronomy, and cryptography.
What's more, calculating pi can spark our interest and passion for mathematics and science, allowing us to appreciate the beauty and charm of mathematics.
epilogue
It is this relentless exploration of pi that allows us to discover more wonders and secrets from the world of numbers. Although only a few digits of pi are needed in many practical applications, the continuous calculation of pi with more digits is not only a scientific exploration, but also a manifestation of human wisdom and technological progress.
Scientists are pushing the boundaries of mathematics and computing in an effort to solve mathematical mysteries and find the patterns behind numbers.
This spirit of exploration and thirst for knowledge is what drives us towards a deeper understanding of the world. Therefore, whether it is for the pure beauty of mathematics or to unravel the mysteries of the universe, it has always been the unremitting pursuit and persistence of scientists to continue to calculate and explore the mysteries of pi.
In this challenging and exploratory process, we may have more unexpected gains, bringing new scientific discoveries and technological advances that will give us a deeper understanding of the world.
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