Why have we been obsessed with solving pi from ancient times to the present?

When it comes to "π", Chinese people will think of Zu Chongzhi, a great scientist during the Six Dynasties period. But many people don't know that Zu Chongzhi was born in Jiangsu, died in Jiangsu, and lived in Jiangsu for most of his life. Many of his scientific achievements, including the calculation of pi, were made in Jiangsu.

Zu Chongzhi calculated the value of pi to 7 decimal places, a record that remained in the world for nearly a thousand years.

From "Three Squares and One Diameter" to the Art of Cutting a Circle

Circle is a ubiquitous figure that pervades every aspect of human life. Human exploration and thinking about circles naturally extend to the study of pi. Pi is the ratio of the circumference of a circle to its diameter. It is a mysterious irrational number with endless decimals. To this day, humans are still exploring it. The calculation of pi has witnessed the development of ancient Chinese mathematics and also confirmed the glory of ancient Chinese mathematics.

According to existing records, the earliest calculation of pi in my country can be traced back to the book "Zhou Bi Suan Jing". The book was written around the 1st century BC. At that time, the book had already proposed the saying "the circumference of a circle is three times the diameter", that is, the circumference of a circle is three times the diameter. However, the book "Zhou Bi Suan Jing" did not give a clear theoretical basis for the saying "the circumference of a circle is three times the diameter". At that time, pi was just a rough value, which was an empirical understanding developed by people in the measurement of acres of land. This statement reflects the initial exploration of pi by ancient Chinese people.

Fang Meiling, director of the History Department of Beijing Normal University, believes that "celestial bodies such as the sun and the moon move along elliptical orbits, and the accurate calculation of pi plays a decisive role in grasping the laws of these celestial bodies and thus accurately calculating the calendar." With the development of society, the "three-dimensional diameter-one" method gradually failed to meet the needs of an accurate calendar. In the 1st century AD, Liu Xin, a scholar of the Western Han Dynasty, created a more accurate pi measurement tool - the Lujia Meter. Liu Xin was the first person to break the "three-dimensional diameter-one" method that had been used since ancient times and pursue a more accurate pi value. He accurately calculated the value of pi to 3.1547, which is known as the Xin method.

From "the circumference of the circle is equal to the diameter of the circle" to the Xin rate, it is not difficult to see that people's calculation of pi at that time was still in the actual measurement, lacking theoretical calculation. This situation was not broken until the appearance of mathematician Zhang Heng in the Eastern Han Dynasty. Zhang Heng calculated pi from the relationship between the circle and its circumscribed square. Zhang Heng's method of calculating pi is recorded in "Suanwanglun" and "Lingxian". Zhang Heng's calculation process is relatively complicated. When Liu Hui, a mathematician in the Wei and Jin Dynasties, introduced Zhang Heng's calculation work, he criticized it and said: "However, the increase in circumference is too much, which is an exaggeration." But it is undeniable that Zhang Heng opened up a new idea and provided a theoretical method for the calculation of pi.

Liu Hui further explored pi and created a method for dividing a circle. In his notes to Nine Chapters on the Mathematical Art, Liu Hui mentioned: "The finer the division, the less the loss. If you divide it again and again until it cannot be divided, it will merge with the circle without loss." In simple terms, it is to divide the circle with a regular polygon inscribed in the circle, and through continuous division, the circumference of the regular polygon is close to the circumference of the circle. The more divisions, the more accurate it is. Liu Hui's method for dividing a circle embodies a kind of extreme thinking and established relevant theories and algorithms for the calculation of pi.

From Zhoubi Suanjing to Jiuzhang Suanshu, from "the circumference of the circumference is equal to the diameter of the circumference" to the method of dividing a circle, we can see the continuous progress of ancient Chinese mathematics in the ancients' constant pursuit of accurate calculations of pi. Empirical knowledge, improvements in measuring tools, theoretical calculations, the emergence of precise scientific methods... The continuous progress of pi calculation methods has promoted the development of ancient Chinese mathematics.

Pi to 7 decimal places

When we talk about pi, we often think of 3.1415926. As early as 480 AD, Zu Chongzhi, an ancient Chinese mathematician, calculated this value. He calculated pi to 7 decimal places and defined the value between 3.1415926 and 3.1415927. This result was not only the most accurate value at the time, but also remained unsurpassed for nearly a thousand years.

There is still no definite answer as to how Zu Chongzhi calculated pi. Zu Chongzhi introduced the method of finding the value of pi in "Zhuishu", but unfortunately "Zhuishu" was lost in the war and has not been passed down to this day. We can only see relevant records in "Sui Shu·Lüli Zhi". "Sui Shu" records: "At the end of the Song Dynasty, Zu Chongzhi, a historian in Southern Xuzhou, developed a secret method, taking the diameter of a circle as 100 million as one zhang, the excess number of the circumference of the circle as 3 zhang 1 chi 4 cun 1 fen 5 li 9 hao 2 sec 7 hu, and the deficiency number as 3 zhang 1 chi 4 cun 1 fen 5 li 9 hao 2 sec 6 hu, and the positive number is between the excess and deficiency. The exact ratio is 113 for the diameter of the circle and 355 for the circumference of the circle. The approximate ratio is 7 for the diameter of the circle and 22 for the circumference."

Although the specific method of calculating pi by Zu Chongzhi was not recorded in the Book of Sui, the results of Zu Chongzhi's calculation of pi were recorded, and the interval, precision and approximate ratio of pi were described in detail. In the view of Fang Yuanxia, ​​a professor at Liaocheng University, if this result was calculated by Zu Chongzhi using the method of dividing the circle, it would require more than 130 various operations on 9 digits, which is undoubtedly a big project. Here we have to mention Zu Chongzhi's main tool for calculating pi - counting rods. Each calculation of pi is a calculation of 9 digits, and it is very difficult to calculate using counting rods. Despite this, the error of Zu Chongzhi's final calculation result is extremely small. Academician Zhang Jingzhong pointed out in the book "The Eyes of Mathematicians" that the error between Zu Chongzhi's precision ratio value of pi and the exact value of π does not exceed 0.000000267.

Zu Chongzhi was an outstanding mathematician during the Northern and Southern Dynasties, and he made many achievements in mathematics. Although his mathematical work "Zhuishu" has been lost, we can still get a glimpse of its style from other records. "The Book of Sui" commented: "No academic official could study its profoundness, so it was abandoned and ignored." People at that time believed that the mathematical theories recorded in "Zhuishu" were very profound and mysterious, and even people with high knowledge could hardly understand them.

The increasing accuracy of the numerical calculation results of pi reflects the progress of ancient Chinese mathematicians in mathematical thinking and methods. It is precisely because of the research of these mathematicians from generation to generation that ancient Chinese mathematics can lead the world.

Extending ideas to modern scientific research

The calculation method of pi in ancient my country contains many subtle ideas, which still influence the development of modern mathematics. In Liu Hui's method of cutting a circle, the more circles are cut, the closer the inscribed polygon is to the circle, and the more accurate the measured data is. This kind of limit thinking was not only an amazing innovative thinking at the time, but also plays an important role in current mathematical research. Limit thinking is an endless calculation process. Modern calculus, numerical analysis, complex variable theory and other mathematical research have all used limit thinking. In addition, the geometric model that plays an important role in pi also occupies an important position in the development of modern mathematics.

Sun Xiaoli, a professor at the Center for Science and Society Research at Peking University, believes that Zu Chongzhi's achievements in pi are inseparable from his scientific spirit of "searching for ancient and modern knowledge", "verifying everything", and never "falsely citing the ancients", and are inseparable from his arduous scientific work and indomitable perseverance. In today's society, innovative development requires "verifying everything", and we must treat scientific research with a rigorous attitude and make unremitting efforts.

Humans are obsessed with pi, and are still calculating it after the decimal point. Using computer technology, modern researchers can calculate pi to trillions of digits. Currently, the precise calculation of pi is mainly for computer performance testing. The more accurate the value obtained by the computer, the stronger the computing power. The increasing accuracy of pi also reflects the continuous progress of modern science and technology.

Sun Yue, Yao Doudou