It has no end and no real numerical value, but if you want to calculate the circumference of a circle that surrounds the entire universe, it can help you. What is the secret of Faye's magic, which makes so many scientists ponder it?
By Ran Levy
Alexander Henry Rynd was not a mathematician. He was a Scottish lawyer, a typical young gentleman of the mid-19th century, with a special fondness for ancient Egyptian culture. Rind suffered from a severe lung disease, and his doctors advised him to stay in a dry climate. For Rind, this was an excellent reason to cross the Mediterranean to the south. During one of his wanderings in the busy markets of the city of Luxor, Rind chanced upon an antique stall and noticed a large sheet of papyrus about six meters wide. Such papyri, often stolen from archaeological sites, have occasionally surfaced in markets. Rind examined the papyrus carefully and decided to purchase it. Unknowingly, the young lawyer purchased at that moment an entrance ticket to the pages of history. The Rhind papyrus contains, as we later discovered, the earliest known value of the most famous mathematical constant: Pi π.
Rind did not get to enjoy his fame because his illness overwhelmed him when he was only about 30 years old, but the papyrus he acquired was studied very thoroughly over the years. The findings indicate that "Papyrus Rind" was written about 1,700 years BC, and is itself a copy of an older papyrus, probably written 300 years earlier. The value of Pi as determined in the ancient document is 3.16, only one percent away from its true value known to us today. As the "Papyrus Rind" testifies, the ancient Egyptians, and also the Babylonians before them, noticed a strange and fascinating feature of circles: if you measure the circumference of the circle and divide it by its diameter, you will get a constant number. It doesn't matter if the circle is as small as a pretzel, or as big as the city wall: the result of dividing the circumference by the diameter will always be the same number, Pi.
What is the secret of Faye's appeal?
Archimedes of Syracuse was the first to succeed in applying the geometrical principles for the purpose of Pi's calculation. He drew a circle, and around it two equilateral polygons: one inside the circle and the other outside it. The perimeter and diameter of the polygons were easy to calculate using simple geometry, and Archimedes proved that the two constitute a lower and an upper limit to the circumference of the circle, which is enclosed between them. In this way, Archimedes came to the conclusion that Pi is approximately 3.14, although even Archimedes knew that this is not the true or final value of this constant.
Evidence of the importance of Archimedes' breakthrough can be found in the fact that for more than 1,500 years no one has been able to calculate Pi with higher accuracy. Archimedes' record held for 1,500 years, and then in just 200 years scientists were able to calculate Pi up to the 100th digit after the point. But Belle was tempted to think that the task had become easy. It still required enormous efforts on the part of the mathematician, who decided to take on the heavy burden of calculating Pi.
Ludolph van Kolen spent most of his life calculating pi up to the 35th digit after the dot. He was so proud of his achievement, which was the best of the 17th century, that he asked to have Fay's value engraved on his tombstone. Why did the mathematicians bother to calculate Pi? What purpose is there in chasing a number that seems to have no end? After all, there is no practical use in knowing the value of pi up to the 100th digit after the point and beyond.
Early mathematicians had good reason to try to calculate phi as accurately as possible. The ancient economy was largely based on agriculture, and the calculation of the cultivated areas (whose borders were not always straight as a ruler) and the length of the winding irrigation canals, were of crucial importance for the farmers. But the exact calculation of Pi was a difficult problem for the Egyptians and their predecessors, since it is not a whole number, but a fraction: three and a bit. In the absence of the necessary mathematical knowledge, they could only use measurements that were actually made for the purpose, measurements that were naturally crude and imprecise.
The intellectual successors of the Egyptians, the Greeks, also had good reasons to calculate Phi. Pythagoras, Euclid and their friends were engaged in the solution of an age-old riddle, the roots of which lie in the mists of history: the "Squaring the Circle" riddle. The question that plagued the Greek philosophers was: Is it possible to draw a square whose area is equal to the area of a circle? The problem is that in order to draw a square whose area is the same as the area of a circle, you need to know exactly the area of the circle. This area is given according to the formula "Pi multiplied by the square of the radius", meaning, we must find out the value of Pi.
For the sake of demonstration, if we wanted to calculate the circumference of a circle, which encircles the entire universe, Pi's accuracy up to the 39th digit after the dot would suffice. Pi's centrality made it a mythical concept and some mathematicians wanted to find out if there was a certain regularity hidden in Pi's apparently random numbers. Such regularity, if it exists, may prove to be a clue to deeper insights into the universe around us.
Simple, too simple
But there were also those who looked for a detour. In 1897, a local doctor, who was also an amateur mathematician, Edwin Goodwin, addressed the members of the General Assembly of the state of Indiana in the USA. He reported to them that he had succeeded in solving the famous "square the circle" puzzle. Goodwin's solution was quite simple: he decided that the value of Pi was 3.2 and that was it. When the value of Pi is clear and known, there is no problem to draw a square with the same area as a circle: calculate the square of the radius of the circle and multiply by 3.2.
Goodwin suggested enshrining his solution in state laws. Members of the Indiana General Assembly referred the bill to the Irrigation Canal Planning Committee (an obvious and logical choice), whose members had enough sense in their heads to refer the matter to the Education Committee. The committee did not find any reason to oppose the determination of Fay's value, since "its current value is so complicated and convoluted that it is not useful at all." From there, the bill went to the state's general assembly, and passed unanimously with zero dissenters. That's when the bill was moved up to the Indiana Senate for final approval before being put on the state statute book.
On the night of the vote on the approval of the law, Professor Clarence Waldo, a professional mathematician from the local university, came to the Senate building to personally oversee the budget of his institution. Someone thrust the bill into his hands and suggested that he go over and congratulate the lucky inventor. Waldo read the bill, realized that it was complete nonsense and at the last minute managed to convince the members of the Senate to shelve the stupid idea.
irrational and transcendental
The first nail in the coffin of the square of the circle puzzle was driven in 1761 when Johann Lambert - a prolific Swiss mathematician who contributed greatly to the fields of astronomy and optics - succeeded in proving that pi is not a rational number. A rational number is a number that can be represented as a fraction. For example, five eighths or a quarter. If Pi cannot be written as a fraction, as Lambert proved, then it is infinite: the digits after the dot go on and on ad infinitum.
The death certificate for the circle square puzzle came about a hundred years later, in 1882, when the German mathematician Ferdinand von Lindemann proved that pi is a transcendental number. A transcendental number is a number that cannot be reached by the conventional methods of addition, subtraction, multiplication or division. Meaning, it is impossible to take any number and from it arrive, through calculation, at the true value of Pi. The deeper meaning is that the true value of Pi cannot be defined. Try as we might, we'll pore over the notebooks and discover new formulas - we'll never reach the true numerical value of Pi, simply because we can't define one.
But there is another type of Fay explorers, who take it elsewhere. They are called "piepologists" and they compete with each other in memorizing the value of pi up to the maximum number of digits. The world record, as of today, belongs to a Japanese, who memorizes the value of pi up to 100 thousand digits after the dot. The famous American Jewish physicist, Richard Feynman, noticed the fact that somewhere in the 762nd place after the dot there is a sequence of six nines one after the other. In one of his lectures, Feynman said that he was interested in learning by heart all the digits, up to the 762nd place, just so he could read them out loud and then finish with "nine-nine-nine-nine-nine-nine". Physicists have a special sense of humor.
The full article was published in the November issue of "Odysseus - A Journey Between Ideas"
Ran Levy is a science writer and hosts the podcastMaking history!', on science, technology and history
More on the subject on the science website
18 תגובות
The point is that Pi is a transcendental number in general and such numbers are used a lot. If I draw a square whose side is 1 (in some units) then I have already drawn a segment with a transcendental length of root length 2 which is the diagonal. without doing anything Therefore, there is nothing problematic about Pi other than the fact that the ancients did not understand that there are numbers that are not whole fractions.
Michael, don't go in there.
What's this nonsense?
Can you provide a link so we can see what you read and didn't understand?
Since it is agreed that pi is the name of the ratio number between the circumference of the circle and its diameter,
And since the circumference of the circle and its diameter constitute a combination of natural length measurements,
Everything we know about this combination of dimensions, amounts to a single knowledge
The circumference is greater than the diameter
This distinction precludes any possibility of knowing the number of the ratio between the circumference and the diameter.
A. Asbar
Note: This topic has been discussed extensively in the math forum.
The area puzzle is the same for a square and a circle. Solution: A string is tied.
Also an exponential exponent is an infinite number about 2.71
I don't know when Feynman said this, but the quote is kind of misleading - what Feynman really said was that he wanted to end with "nine, nine, nine, nine, nine, nine and so on" (And so on), which means that from here on the number continues with only nine, so it is rational. Somehow it seems to me that this kind of joke (which requires some mathematical knowledge to understand) is more suitable for Feynman than an obscure mention of the Beatles.
Most likely it has nothing to do with the humor of physicists, assuming that it was said after the year sixty-eight, this is probably a reference to the passage Revolution 9 of the Beatles.
The article is over a year old:
http://www.ynet.co.il/articles/0,7340,L-3630050,00.html
Oops. Not Yuhai. Devil's advocate, of course.
Yohai - of course I understand. Therefore, I hope you will trust me that if, despite this understanding, I believe that this is a principled and essential point and not a "pettiness", there is something behind it.
In this case, I point out that the prism through which Ren looks at the whole concept of "defined" and "calculable" is irrelevant and immaterial. Ren limits the meaning of the calculation and representation of a number in a very, very arbitrary manner (he only allows the final activation of the four calculation operations and demands that the number be written in its entirety on paper) and there is no logic or reason behind these limitations unless you look through the prism of a limited person living a limited life , and then comes into the picture the monstrous initial I gave, which is even less obvious than pie.
In essence, pie is defined as beauty, and beauty can be calculated. Any digit of Pi that we want, we can know what it is after a finite amount of time - just like any "real" number written on paper, the meaning of this for us is that we can know its every digit after a finite amount of time (after all, we cannot "capture" a huge natural number in its entirety, even if it is all written on paper). In my opinion, there is no fundamental difference between the numbers on this point, which means that I do not agree with the *main principle point* that Ran presents in the article (mathematically what he writes is simply wrong - his reasoning for why the circle cannot be quartered is incorrect - but such mistakes are easier to forgive ).
I really appreciate Ran and his attempt to transfer knowledge to as large a section of the public as possible. Most of this article does that job well. The problem begins when the article tries to do something beyond the transmission of knowledge - to invent interpretations and meanings for this knowledge that do not exist. This is something that is not true and should not be passed on to the public. The Rhind papyrus and Indiana's Pye Act can suffice.
Yochai
How do you draw a side whose length is pi? When exactly do you lift the pencil from the page?
An estimate of the radius of the universe is about 10 to the power of 26 meters. If our instruments today can measure up to an angstrom, about 36 is needed after the point to estimate how many angstroms surround the universe. If we assume that the devices are a bit better than that, and the universe is a bit bigger, then 39 is a reasonable number.
Yochi: That's right, the smallest number is the best we know how to measure and the largest is the best estimate we have for the size of the universe.
Capricorn, you're a bit trifling, the prime number that you describe, we can't write for technical reasons, we can, on the other hand, recite it if we live long enough, we can't write Pi for conceptual reasons, you surely understand that. Even if the first monkey who learned to talk started reading the number and the last person to die in the apocalypse finished it would not be even a fraction of the number. But again, I'm sure you understand. Regarding wording: according to my (positive) impression of Ran's blog, his goal (as mentioned, in my opinion) is to transfer knowledge to as large a section of the public as possible and not only to people from the field who already understand and probably know it, the way they word for the different audiences is different.
Getting caught up in subtle and "small" formulations will not help convey the knowledge to a wide public.
There is actually no problem in writing the whole thing on a piece of paper, and I have done it many times.
I join Capricorn in rebellion, but I'm a little more blunt.
The article is full of inaccuracies that are not mathematical pettiness, but indicate a lack of understanding.
-"If Pi cannot be written as a fraction, as Lambert proved, then it is infinite: the digits after the dot go on and on ad infinitum." This is also true for rational numbers, such as a third.
-"For the sake of demonstration, if we wanted to calculate the circumference of a circle, which surrounds the entire universe, Pi's exactness up to the 39th digit after the dot would suffice." What is the meaning of this sentence? why 39? What is the size of the universe?
- It is very easy to draw a square whose area is the area of a circle. Just draw a square whose side is the length of the Pi root. Only you can't do it with a ruler and a caliper.
Regarding the problem of squaring the circle, it is a slightly more complicated problem than described. This is a construction in which it is allowed to use only a compass and a ruler (one that draws straight lines, but you cannot measure distance with it). A related problem is dividing an angle into 3 equal parts. Although no transcendental numbers are involved here, it has already been proven that this is an impossible construction.
Unfortunately I have to rebel here - the statement "we will never reach the true numerical value of Pi, simply because we cannot define one" is simply wrong; Wrong use of the word "set". Determining the true numerical value of pi is possible without difficulty. Regarding "arriving" - a concept that is not completely clear - it may be true that we will never be able to write Pi in its entirety on a sheet of paper, but we will also not be able to write the Googleflex prime number in its number on paper because there are not enough atoms in the universe (but I don't think there is an argument that the natural numbers have a "real numerical value").
Another nitpick: "A transcendental number is a number that cannot be reached by the conventional methods of addition, subtraction, multiplication or division." - This definition is also incomplete (because it does not say where one "starts" the attempt to reach), and also describes an irrational number, if we make the reasonable assumption that one starts from natural numbers - that is, it is not a description of the property of transcendentality (the algebraic root of 2 is also impossible arrive with addition, subtraction, multiplication or division if starting from the natural ones). Unfortunately I have already pointed out this error before when I read the article elsewhere and it has not yet been corrected.
In your example about the number of digits needed to know a fraction to encompass the universe, you also need to specify the smallest size - for example to be able to distinguish a football. The precision is a relative number between the maximum size and the minimum size.
http://www.smbc-comics.com/index.php?db=comics&id=1809