- 3433 - “Pi’ - how can 3.14.….go on forever? “Pi“, the ratio of a circle's circumference to its diameter, is one of the best-loved and most interesting numbers in math. Like "e," it seems to suddenly arise in a huge number of math and physics formulas. What Makes Pi so special?
------------- 3433 - “Pi’ - how can 3.14.….go on forever?
- Mathematics is one of the only areas of knowledge that can objectively be described as "true," because its theorems are derived from pure logic. And yet, at the same time, those theorems are often extremely strange and counter-intuitive. Politics is the latter but not the former.
-
- Random numbers are hard to come by. Random data isn't actually all that random. In a given list of numbers representing anything from stock prices to city populations to the heights of buildings to the lengths of rivers, about 30 percent of the numbers will begin with the digit 1.
-
- Less of them will begin with 2, even less with 3, and so on, until only one number in twenty will begin with a 9. The bigger the data set, and the more orders of magnitude it spans, the more strongly this pattern emerges. What’s with this?
-
- Because prime numbers are indivisible (except by 1 and themselves), and because all other numbers can be written as multiples of them, prime numbers are often regarded as the "atoms" of the math world.
-
- Despite their importance, the “distribution” of prime numbers among the integers is still a mystery. There is no pattern dictating which numbers will be prime or how far apart successive primes will be.
-
- In 1963, the mathematician Stanislaw Ulam noticed an odd pattern while doodling in his notebook during a presentation: When integers are written in a spiral, prime numbers always seem to fall along diagonal lines. This in itself wasn't so surprising, because all prime numbers except for the number 2 are odd, and diagonal lines in integer spirals are alternately odd and even.
-
- Much more startling was the tendency of prime numbers to lie on some diagonals more than others and this happens regardless of whether you start with 1 in the middle, or any other number.
-
- Even when you zoom out to a much larger scale, as in the plot of hundreds of numbers , you can see clear diagonal lines of primes , with some lines stronger than others. There are mathematical conjectures as to why this prime pattern emerges, but nothing has been proven. Something for you to work on!
-
- In an important field of mathematics is called “topology“. Two objects are considered to be equivalent, or "homeomorphic," if one can be morphed into the other by simply twisting and stretching its surface; they are different if you have to cut or crease the surface of one to reshape it into the form of the other.
-
- Consider a torus, the dougnut-shape object. If you turn it upright, widen one side and indent the top of that side, you then have a cylindrical object with a handle. Thus, a classic math joke is to say that topologists can't tell their doughnuts from their coffee cups.
-
- On the other hand, “Moebius bands“, are loops with a single twist in them, and are not homeomorphic with twist-free loops (cylinders), because you can't take the twist out of a Moebius band without cutting it, flipping over one of the edges, and reattaching.
-
- Topologists long wondered: Is a sphere homeomorphic with the inside-out version of itself? In other words, can you turn a sphere inside out? At first it seems impossible, because you aren't allowed to poke a hole in the sphere and pull out the inside. But in fact, "sphere eversion," as it's called, is possible.
-
- There's just a finite number of distinct geometric patterns. All Escher paintings, wallpapers, tile designs and indeed all two-dimensional, repeating arrangements of shapes can be identified as belonging to one or another of the so-called "wallpaper groups." And how many wallpaper groups are there? Exactly 17. How can that e, exactly 17?
-
- The letter "e" represents an irrational number (with unending digits) that begins 2.71828... Discovered in the context of continuously compounded interest, it governs the rate of exponential growth, from that of insect populations to the accumulation of interest to radioactive decay.
-
- In math, the number exhibits some very surprising properties, such as being equal to the sum of the inverse of all factorials from 0 to infinity. Indeed, the constant "e" pervades math, appearing seemingly from nowhere in a vast number of important equations.
-
- Next, "i" represents the so-called "imaginary number": the square root of negative 1. It is thus called because, in reality, there is no number which can be multiplied by itself to produce a negative number (and so negative numbers have no real square roots).
-
- But in math, there are many situations where one is forced to take the square root of a negative. The letter "i" is therefore used as a sort of stand-in to mark places where this was done.
-
- “Pi“, the ratio of a circle's circumference to its diameter, is one of the best-loved and most interesting numbers in math. Like "e," it seems to suddenly arise in a huge number of math and physics formulas. What Makes Pi so special?
-
- Putting it all together, the constant "e" raised to the power of the imaginary "i" multiplied by pi equals -1. And, as seen in Euler's equation, adding 1 to that gives 0. It seems almost unbelievable that all these strange numbers, and even one that isn't real would combine so simply. But it's a proven fact.
-
---------------------------------- e^ix + 1 = 0
-
- I am sure one of my readers will make sense of this and give us an explanation.
-
January 25, 2022 “Pi’ - how can 3.14.….go on forever? 3422
----------------------------------------------------------------------------------------
----- Comments appreciated and Pass it on to whomever is interested. ---
--- Some reviews are at: -------------- http://jdetrick.blogspot.com -----
-- email feedback, corrections, request for copies or Index of all reviews
--- to: ------ jamesdetrick@comcast.net ------ “Jim Detrick” -----------
----------------------------- Thursday, January 27, 2022 ---------------------------
No comments:
Post a Comment