If you ask anyone on the street what they know about pi, the most common answer is that it is a decimal fraction of 3.14. Few will expand on the answer, remembering the 7th grade program: “This is the ratio of the circumference of a circle to its diameter” or “This is a decimal fraction – 3.14 … which has an infinite number of decimal places.”
Let’s clarify – in June 2022, restless mathematicians added the first 100 trillion (!) Signs to it, and this, they believe, is not the limit.
There are a lot of similar “infinite” or irrational numbers, whose decimal representation can last indefinitely, more than rational ones.
— Yuri Valentinovich, why did the number pi get so much attention?
– The number pi is associated with a circle – one of the simplest geometric shapes that often occurs in our lives, and therefore it appears in any area where periodic processes occur. And this is astronomy, where, for example, you need to calculate the orbits of celestial bodies, artificial satellites, the trajectories of rockets, architecture and electrical engineering, physics, electronics, chemistry, navigation, mathematics and other areas.
Why is the Greek letter pi used to represent this number?
– The Greek word begins with it, which in translation into Russian means “periphery, circle”. So the letter was chosen to express the ratio of the circumference of a circle to the length of its diameter.
The great scientist Leonhard Euler used this designation in many of his works. It turned out to be convenient, took root in mathematics, and from there it passed into our lives. For any circle, large or small, this ratio is the same. Its approximate value is 3.1415926 … The ellipsis put here means that a number of numbers can be written after the number 6. Together with the written ones, they will give a more accurate approximate value of the number. This series of numbers can be continued arbitrarily far.
– I heard that in 2022 the first 100 trillion digits of pi after the decimal point were calculated …
– This is true. It would take more than 3.1 million years to read them all aloud, one per second. And the one hundred trillionth decimal place of pi is zero. We can get as close to pi as we like, but we will never be able to get its exact value in this way. As they say, the decimal representation of pi is infinite. It can be said in another way: the circumference of a circle of unit diameter can only be measured approximately.
Help “MK”. Numbers equal to the ratio of two integers are called rational, and all other numbers are called irrational. Rational numbers correspond to finite decimal fractions or infinite but periodic fractions. At the same time, there are much more irrational numbers than rational ones. They can’t even be counted.
– Tell us about the historical roots of the number pi.
“I will tell you about an old problem that has been waiting for its solution for more than two thousand years. We are talking about measuring the area of a circle.
For the ancient Greeks, the words “measure the area of a figure” meant: to build a square with the same area as this figure with the help of a compass and a ruler without divisions. They learned to do this for triangles and rectangles, for any polygons in general, for some curvilinear figures. But it didn’t work for the circle. The problem got its own name – “squaring the circle”, and in attempts to find it, good approximations of pi to rational numbers were found.
For example, the ancient Greeks believed that the circumference was equal to 22/7 of the diameter, and this, as we now know, approximate equality fully provided for their needs, say, in the construction business. If we represent the number 22/7 as a decimal fraction, then we will see an infinite series, it is also periodic: 22/7 u003d 3.142857142857 …, the combination “142857” is repeated in it an infinite number of times. Note that the first two digits after the decimal point for the fraction 22/7 and for the number pi are the same. This means that the fraction 22/7 approximates well the ratio of a circle’s circumference to its diameter.
And in Babylon, an even more accurate approximation was known: 355/113 = 3.141592 … much more accurate than 22/7.
In general, the task of finding the squaring of a circle was very attractive, it had a simple and understandable formulation, a venerable age, and, despite considerable efforts, was inaccessible to many professionals and amateurs. Only in 1882 did the German mathematician Ferdinand Lindemann manage to prove that there is no construction realizing the squaring of the circle, that the squaring of the circle is impossible.
– Can you give another example of non-measurable geometric objects?
– There are a lot of them. For example, the diagonal of a square and its side are incommensurable. This fact was discovered by ancient Greek scientists. The length of the diagonal can only be measured approximately.
Let’s take a meter ruler with risks marked on it at a distance of one millimeter and try to measure the length of the diagonal of a square with a side of 1 meter with it. If we put a meter on the diagonal, and then try to measure the rest of it with a ruler, then the end of the diagonal will fall between the little lines. It is possible to divide the ruler into smaller parts, and again none of the marks of the new division will coincide with the end of the diagonal. The end of the diagonal will always fall between two adjacent risks, no matter how small the division you make. The ratio of the lengths of a side of a square and its diagonal is an irrational number.
— And who was the first to prove the irrationality of pi and how did he do it?
– The irrationality of this number was first proved back in 1761 by Johann Lambert. He used for this the so-called continued fractions, exponential and trigonometric functions.
– If the ancient Greeks were satisfied with the approximate value of pi, modern schoolchildren, students also manage with it, then why do scientists continue to calculate hundreds of trillions of decimal places?
– This question was answered by the head of the group, who counted so many characters: such calculations demonstrate the power of the available computer technology. The calculation of many signs is a kind of sports competition between groups of scientists who create computer technology, come up with more advanced algorithms and computer programs. Of course, this requires a lot of money, but I think not more than drug advertising, for example.
— Are there any other numbers that attract mathematicians so much?
— I don’t know if there is a holiday of the number e… But this is another constant that is no less famous among engineers and scientists than pi. She is also irrational. No one has yet answered the question: do we get a rational number by adding e to pi? This is an old problem that no one can solve.
What does the number e look like?
– You can approximately write it as a decimal fraction 2.7128 … It is also calculated to trillions of decimal places. It has not a geometric, but an analytical origin.
– You are the winner of awards: Hardy-Ramanujan (1997), Humboldt (2003), Markov (2006). For the solution of which theoretical problem have you been awarded several international prizes in a row?
– It is connected with the numbers pi and e. As I said, these are two mathematical constants, but whether there is an algebraic connection between them is an unresolved and very difficult question. I considered the numbers pi and e to the power of pi. It would seem that the number e to the power of pi is more complicated than just the number e, but nevertheless I managed to prove that these numbers are algebraically independent.
— Can they be used in cryptography?
– Sometimes for computer calculations there is a need to build sequences of random numbers. This is necessary for many tasks, including cryptography. There is an assumption that the digits of the decimal fraction of pi are located randomly. This sequence of numbers does not have a period, but it is possible that there are other ratios unknown to us so far. This is a hypothesis that has not been proven or refuted.
– Are there any other difficult, unsolved mathematical problems related to the number pi?
– Of course they do. For example, it is not known whether each digit from 0 to 9 occurs in decimal pi an infinite number of times. And if so, which number is more common? Maybe, on average, all numbers appear equally often? Computer calculations support the latter hypothesis, but it is still unproven.
– It is believed that in the number pi everyone can find their phone number, bank account, and so on. This is true?
— This is another of the known open problems. The question, in general, is put like this: is it possible to find any given sequence of digits in the decimal fraction of pi? Answer: it is still unknown – the fraction is infinite. For example, you may or may not find a bank account of 20 known digits. If you don’t find it, wait until the next 100 trillion is calculated, maybe your bank account will be there. (Smiling.)
– Did you personally look for something?
– No, it’s a waste of time. What is it for?
Yes, out of sport.
– Well, except to arrange sports competitions, who will quickly find their phone number or account, and then give out a prize for it. But, in my opinion, proving a hypothesis is a more interesting goal. True, it is almost hopeless.
Do you celebrate this holiday?
– No, he, in fact, is not so many years old for this to become a tradition. In addition, this holiday was born in the United States and is associated with their system of recording dates. In Russia, as, indeed, in many other countries, say, in England, dates are written in the order of day-month-year. And in the US, the order is different – month-day-year. Therefore, March 14 in the United States will be written as 3.14, and in Russia – 14.3. The American notation corresponds to the first three decimal digits of pi, while the Russian one, 14.3, has nothing to do with this number. It turns out that we have nothing to celebrate on March 14th.
– Maybe there are other “mathematical” dates that, in your opinion, are worth celebrating?
– In 1973, at the Department of Number Theory of Moscow State University, we celebrated another event – the century of proof by the French mathematician Charles Hermite of the transcendence of the number e. Transcendence means that this number cannot be the root of any polynomial with integer coefficients.
From this event, in essence, the development of a large area of number theory began, and Russian mathematicians took an active part in this process. Lindemann, trying to prove the impossibility of squaring a circle, proved a much more general statement – the transcendence of pi.