Happy Pi Day!!!
Yes math geeks, it’s March 14th otherwise known as 3.14, the approximation of the number Pi. The number Pi or “π” is the ratio of a circle’s circumference to its diameter. No matter how big or small you make the circle that ratio will remain the same. Pi has been known to the ancient Babylonians and Egyptians although it is through Greek mathematics that Pi has achieved cultural prominence.
π is an irrational number. Irrational numbers cannot be expressed as the ratio of two integers and their decimal expansions are infinite and non-repeating. In other words, the decimal expansion of π goes on forever. The first fifty digits of π are 3.14159265358979323846264338327950288419716939937510.
Using computers, mathematicians have calculated the first 10 trillion digits for π ! Now before you get excited about that accomplishment please stand up, reach to the sky and jump straight up. For that brief moment you are immeasurably closer to nearest galaxy 1 million light years away then we are to finding all of the digits of π. Pi is so huge that no matter how far you go in expanding Pi you are just getting started.
Pi: Transcendental and Irrational
Pi has the extra distinction of not only being Irrational but Transcendental. There are two kinds of Irrational Numbers: plain old Irrational numbers like the square root of 2 and Transcendental Numbers like “e” and “π”. Transcendental Numbers are not algebraic—that is, it is not a root of a non-constant polynomial equation with rational coefficients. Here is an example of a simple polynomial equation: 0= x2 + 2x +5. Because π is Transcendental, you could not calculate it exactly using addition, subtraction, division and multiplication using rational numbers in a finite number of steps. You could use any rational number(s) you want but you will never get there.
Though only a few classes of Transcendental numbers are known (in part because it can be extremely difficult to show that a given number is transcendental), Transcendental numbers are not rare. Indeed, almost all real and complex numbers are Transcendental, since the algebraic numbers are countable while the sets of real and complex numbers are both uncountable.
The most Beautiful Equation in Mathematics
Pi is mysteriously linked to other fundamental mathematical constants in Euler’s brilliant equation:
Here linked together:
– The number 0, the additive identity.
– The number 1, the multiplicative identity.
– The number π
– The number e, the base of natural logarithms, which occurs widely in mathematical and scientific analysis (e = 2.718281828…).
– The number i, the imaginary unit of the complex numbers, a field of numbers that contains the roots of all polynomials (that are not constants), and whose study leads to deeper insights into many areas of algebra and calculus, such as integration in calculus.
Is Pi normally distributed?
Another mystery of π. Imagine a 10 sided die with the digits 0 through 9 on the sides and imagine using this die to determine the next digit of Pi. Roll the die….. a “4”. What was the odds of getting a 4? Why it’s 1/10 because there are 10 possible outcomes. What are the odds of getting two 4’s in a row? It’s 1/10 times 1/10 or 1/100. How about three in a row? 1/10 x 1/10 x 1/10 or 1/1000. It’s possible but not very likely. Now imagine a hundred “0”s in row? Or a million? Or even a trillion… The odds of getting the same number on a trillion consecutive rolls of a die are extremely small. However, If the digits of π are normally distributed like the would be if you rolled a ten sided die than any number sequence you can imagine would eventually show up in Pi if you calculated far enough. Why? Because the occurrence of any event, not matter how small the odds as long as it isn’t zero, is 100% over infinity. No one has proved that this is true for Pi or any irrational number but it does appear to be true. The first 10 trillion digits of Pi look like they are normally distributed though. Some examples are the six “9”s at the Feynman Point (762nd digit) and at the 193,034th place and there is a sequence of 12 zeroes starting at position 1755524129973.
Philosophical implications of Pi.
However, If it the digits of Pi are not normally distributed like we suspect then how are they generated? Would that imply that there is a pattern?
On the other hand, saying that Pi is normal and the digits of Pi are like the random rolls of a ten sided die raises some problematic questions too. Does it imply that the digits of Pi are determined only as mathematicians and their computers calculate them just like the quantum states of sub-atomic particles are determined by our measuring them? If that’s the case, could a highly advanced civilization “force” the digits of Pi into patterns as they calculate them? (Don’t laugh, it’s the premise of Carl Sagan’s novel “Contact”). If this is true it then the implications are stunning: There is a mysterious link between consciousness and the mathematical fabric of the universe.
Looking at it from the other side doesn’t let you off easy either. If the entire infinite set of the digits of Pi are already predetermined then where are they stored? In what fashion does this information exist independent of human thought?
Hmm…Deep questions with even deeper implications for the meaning of human existence and the meaning of truth no matter what direction you go. It’s probably a good thing that St Patrick’s Day is three days after Pi Day!
Post Script: For a fun way to look at the discovery of irrational numbers check out the Fable of Phidas and Tekton