What is the essence of Pi?
I was thinking about numerical sets and suddenly got confused. We know that Pi is an irrational number. We also know that Pi is the ratio of the circumference of a circle to its diameter. That is, we are essentially dividing two numbers. And here it became unclear to me: where do we get the irrationality of Pi, if when dividing rational numbers, we should always get a rational number. It turns out that the length of a circle and its diameter are always irrational numbers? But in this case, how could we get such an exact value of Pi, if with the irrationality of the diameter and length, we could never calculate this value so accurately?
You have made a mistake in assuming that both values are rational.
In fact, only one of them can be rational – either the circumference or the diameter. Either both are irrational. If the diameter is rational 3 cm, then the circumference is 3 * Pi-irrational.
As for calculations-for a long time, mathematicians calculated the number Pi, let's say, geometrically – that is, banally making measurements and some transformations of the drawn figure.
Since the fifteenth century, when mathematics began to study infinite series, they began to study the” distant ” digits of this number (that is, more than ten digits after the decimal point). This was done by trial and error. Some mathematician came up with a formula. A few years passed – another mathematician proved that the previous formula was wrong on the twentieth digit and offered his own, which is accurate on the twentieth digit. Only in the eighteenth century did mathematics develop to the point where it was able to prove a number of formulas (namely, formulas related to infinite series) that involve the number Pi (and from which it can be calculated).
For example, the Basel problem solved by the greatest Leonhard Euler.
From it, you can calculate exactly the value of Pi with any required accuracy. If you spend a couple of years or more with a pen in your hand doing calculations. With the advent of computers, the question generally moved to the stage of “find the formula that takes up the least amount of RAM” – a thousand characters? a million? a billion? Yes, no question, give a more powerful computer and everything will be fine. On personal computers, for example, the Chudnovsky brothers formula is used:
But in general, now the number Pi is calculated, which is called “for show-offs” – just to check the computing power of the computer, for example. There is absolutely no practical value in writing at least fifty decimal places.
What about the number e? The fact is that e to the power of x, when determining the derivative of this function, the derivative does not change. This is the only such number. A little more, or a little less and collapse.
It's the same with the number “Pi”. This is the ratio… at which space does not collapse.
By the way, and about the number, in general. It is contained, 1st-by the place in the sequence, 2nd-by the direction of this sequence, 3rd-by the metric ratio of the proximity of elements of this axis.
A number is a minimum three-dimensional object (by the way, there is no less than one).
But it is postulated, from and what crawls out, and God, and the devil in one person (and economism with Marx-Lenin, in the same place Hitler with Stalin and Izrailevka…) and all this is called logic (they say you will not go anywhere).
Pi is a constant similar to Planck's constant, the gravitational constant, and so on. But so far, no one has answered the question of what it is… I would very much like to hear a competent opinion on this issue.
What kind of entity are you asking about? If we are talking about mathematics, then it has been known since Ancient Egypt, if not earlier: the ratio of the circumference of a circle on a plane to twice its radius or diameter.
If you are interested in the heuristic essence of π, then all 10 digits (from 3 to 0 inclusive) appear in it during 33 steps(together in the first, integer 3). Here is this curious series: 3,1,4,5,9,2,6,8,7,0. The structure in it is visible to the naked eye, but what is its true meaning? First 3, then 1, then Fibonacci: 3+1 = 4, 1+4 = 5, 4+5 = 9. The first half is lined up. The second part is somewhat trickier to create: 2 × 7 = 14 and 6+8 = 14. Finally, the difference 14-14 = 0, the calculation is finished. There are three arithmetic operations: addition, multiplication, and subtraction. But you can also find the division: 6+8 = 14, then 14:2 = 7. It is clear that these are just “mind games” (the funny thing is that my last name, if the letters of the alphabet are numbered from 1 to 33, gives exactly the same numbers as the word “mind”!😁).
Finally, there is a conspiracy version of the number π, which deduces cosmological information from its transcendental construction. But I don't think this is reasonable at all.😉
However, for the Truth with a capital letter, my opinion may not mean anything…