Russian Academy of Sciences member Yuri Nesterenko: “Your phone and your bank account can be hidden in it”
The night is approaching, when some citizens bake round pies decorated with numbers, and then they cling to the clock and carefully watch the hands. It is important not to miss their cherished combination with the date: 3.14 1:59:26. All this for the sake of the «magic» number pi, which they write down to the seventh decimal place — 3.1415926, and on March 14 at 1 hour 59 minutes 26 seconds they celebrate the birthday of the number pi.
About old and new problems related to pi, and unexpected surprises associated with it, we talked with Yuri Nesterenko, winner of several international awards in the field of number theory, head. Department of Moscow State University. M.V. Lomonosov, Corresponding Member of the Russian Academy of Sciences.
If you ask anyone on the street what they know about pi, the most common answer is that it's 3.14 decimal. 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 — 3.14… which has an infinite number of decimal places».
Clarify — by June 2022, restless mathematicians have calculated the first 100 trillion (!) decimal places … and this, they believe, is not the limit.
There are a lot of similar irrational numbers whose decimal representation can last indefinitely , more than rational ones.
— The number pi is associated with the circle — one of the simplest geometric figures that is often found 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, rocket trajectories, architecture and electrical engineering, physics, electronics, chemistry, navigation, mathematics and other areas.
< p>— The Greek word begins with it, which in translation into Russian means «periphery, circle». Here is the letter 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. You can continue this series of numbers as far as you like.
— This is true. It would take more than 3.1 million years to read them all aloud, one per second. And one hundred trillion decimal place of pi — 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.
Numbers equal to the ratio of two integers are called rational, and all other numbers — 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. You can't even count them.
— I will tell you about an old problem that has been waiting for its solution for more than two thousand years. It's about measuring the area of a circle.
For the ancient Greeks, the words «measure the area of u200bu200ba figure» meant: to construct with the help of a compass and a ruler without divisions a square having the same area as this figure. They learned how 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 task got its own name — «squaring the circle», and in trying 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 = 3.142857142857…, the combination «142857» repeats itself 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 problem of finding the squaring of the circle was very attractive, it had a simple and understandable formulation, a respectable 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, the squaring of the circle is impossible.
— 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 — the number is irrational.
— The irrationality of this number was first proved back in 1761 by Johann Lambert. He used the so-called continued fractions, exponential and trigonometric functions for this.
— 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. Calculation of many characters — it 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.
— I don't know if there is a holiday for 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.
— You can roughly write it as a decimal fraction of 2.7128… This is also calculated to trillions of decimal places. It has not a geometric, but an analytical origin.
— It is connected with the numbers pi and e. As I said, these are two mathematical constants, but is there an algebraic connection between them — the question is unresolved and very difficult. 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.
— Sometimes for computer calculations it becomes necessary 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.
— 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 confirm the last hypothesis, but it is still not proven.
— This is another of the known open issues. The question, in general, is put like this: is it possible to find any given finite 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.)
— No, it's a waste of time. What is it for?
— 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, to prove the hypothesis — more interesting target. True, it is almost hopeless.
— 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 USA the order is different — month-day-year. Therefore, March 14 in the USA will be written as 3.14, and in Russia — 14.3. The American notation corresponds to the first three decimal digits of pi, and the Russian — 14.3 — has nothing to do with this number. It turns out that we have nothing to celebrate on March 14.
— In 1973, at our Department of Number Theory of Moscow State University, we celebrated another event — a 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 direction in 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 — transcendence of pi.