Mathematics is absolutely beautiful and here I have just tried to capture some of its beauty. I hope you enjoy reading this!
• Put a map of your country on the floor, there’s a point on the map that is touching the actual point it refers to. This is a cool instance of the Banach fixed-point theorem.
• A spin-off to the Borsuk-Ulam theorem states that at a given instant, there exists at least one pair of antipodal points (the antipodal point of a point on the surface of a sphere is the point which is diametrically opposite to it) on the earth which have the same temperature and pressure.
• The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3-dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. A stronger form of the theorem implies that given any two "reasonable" solid objects (such as a small ball and a huge ball), the cut pieces of either one can be reassembled into the other. This is often stated informally as "a pea can be chopped up and reassembled into the Sun" and called the "pea and the Sun paradox".
• Every simple closed curve that you can draw by hand will pass through the corners of some square. This question was asked by Toeplitz in 1911 and remains unsolved to this day.
• In 1736 the mathematician Leonhard Euler was trying to find a way to cross every single one of the seven bridges of Königsberg exactly once. He realized that this seemingly banal problem was impossible to solve and in the process, he created a new area of mathematics, Graph Theory.
• Pascal's Triangle as a prime number tester - Pascal's Triangle has a very interesting property related to prime numbers. The lead prime number in a horizontal row divides evenly into all the other numbers in that row except for ‘1’. This does not hold true for composite numbers. Pascal's Triangle, itself, is a legitimate primality test. For example, ‘5’ divides evenly into ‘10’, ‘10’ and ‘5’.
• Surprisingly, shading in the odd numbers of Pascal’s Triangle reveals the infinite Serpienski fractal.
• A seemingly elementary problem that no one has solved: the Singmaster's Conjecture - There's a number A so that no value occurs in Pascal’s triangle more than A times.
• Any number dialed in a rectangular shape on a calculator numpad is a multiple of 11.
• 1729 is known as the Hardy-Ramanujan number, after an anecdote of G. H. Hardy when he visited Srinivasa Ramanujan in hospital where Ramanujan said "It is a very interesting number; it is the smallest number expressible as the sum of two (positive) cubes in two different ways. ()” He could instantly tell that because he was actually trying to solve Fermat’s Last Theorem. is a near miss solution to Fermat’s Last Theorem, as it only misses by 1 (or ). Ramanujan identified a way to generate an infinite number of near misses of the form
• The Collatz conjecture or 3n+1 conjecture is a conjecture in mathematics that concerns a sequence defined as follows: start with any positive integer n. Then each term is obtained from the previous term as follows: if the previous term is even, the next term is one half the previous term. If the previous term is odd, the next term is 3 times the previous term plus 1. The conjecture is that no matter what value of n, the sequence will always reach 1. For instance, starting with n = 12, one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1.
• Goldbach's conjecture is one of the oldest and best-known unsolved problems in number theory and all of mathematics. It states: Every even integer greater than 2 can be expressed as the sum of two primes. A modern version of Goldbach's marginal conjecture is: Every integer greater than 5 can be written as the sum of three primes.
• A not so famous Goldbach Conjecture - Goldbach stated that every positive odd integer is either prime or the sum of a prime and twice a square. 5777 and 5993 are the only known counterexamples.
• Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu. The integers and are two powers of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive powers. That is to say, that the only solution in the natural numbers for a, b > 1, x, y > 0 is x = 3, a = 2, y = 2, b = 3.
• Here's what you get if you divide instead of multiplying in n!
• 1084 is the smallest number whose English name contains all five vowels in order - one thousAnd and EIghty-fOUr.
• 2⁸²⁵⁸⁹⁹³³-1 is the largest known prime number and was discovered on 21st December 2018. It's substantially easier to find primes of the form 2ᵖ-1, also called Mersenne Primes.
• Take a number N:
1) Add its digits
2) Repeat until the result has 1 digit
The total number of iterations is the additive persistence of N. The smallest number in base 10 of additive persistence 4 is 19,999,999,999,999,999,999,999.
1) Add its digits
2) Repeat until the result has 1 digit
The total number of iterations is the additive persistence of N. The smallest number in base 10 of additive persistence 4 is 19,999,999,999,999,999,999,999.
• A curious identity involving e and π
• Here’s why a 20 digit number starting with eleven 1's cannot be a perfect square
• Every time you shuffle a deck of cards, chances are that you have put them in an order that has never been seen in the history of the universe. There are 52! ways to arrange the cards. That’s an extremely large number. Scott Czepiel has come up with a story to explain just how large of a number it is : Start by setting a timer to count down from 52! seconds. Now, choose a spot on the equator and take one step every billion years. Once you have made it around the earth, remove a single drop from the Pacific Ocean and then walk around the Earth again. Continue doing the same thing over and over until the ocean is empty. Once all of that is done, you won’t even have made a dent in the amount of time left on the timer.
• Tupper's self-referential formula is a formula that visually represents itself when graphed at a specific location in the (x, y) plane. The formula is an inequality defined as:
If one graphs the set of points (x, y) in 0 ≤ x < 106 and k ≤ y < k + 17 (k is a 543-digit integer) satisfying the inequality given above, the resulting graph looks like this (the axes in this plot have been reversed, otherwise the picture would be upside-down and mirrored):
• In this supermagic square, not only do the rows, columns and diagonals add up to 34, but so do all the combinations of 4 numbers marked by linked dots in the squares below.
• An amazing pandigital approximation to e that is correct to 18457734525360901453873570 decimal digits is given by, (found by R. Sabey in 2004)
•There isn’t enough room on the entire planet to write out a googolplex. A googol is 1 followed by 100 zeros. A googolplex is 1 followed by a googol zeros. It is so big that if you were to write the whole thing out in books, they would weigh more than our entire planet, far more in fact. They would weigh about kg, whereas the Earth weighs about 5.972 x kilograms.
• There’s no such thing as a boring number in math. We can prove this using what mathematicians call proof by contradiction. Let’s say that there were one or more boring numbers (with no special characteristics), let n be the smallest one. Except now n is interesting, because it is the smallest non-interesting number.
• In probability theory, the birthday problem or birthday paradox concerns the probability that, in a room of people, there’s a % chance that two people have the same birthday. In a group of people, there are pairs of people. The chance of any particular pair having different birthdays is . However, the chance of all pairs having different birthdays is . It may seem counterintuitive, but it’s true. If you increase the number of people in the room to , then you are almost guaranteed (with a % chance) that at least two people have the same birthday.
• According to Buffon's needle problem, if we have a floor made of parallel strips of wood, each the same width, (i.e. a floor ruled with equidistant parallel lines) and we drop a needle onto the floor, the probability that the needle will land on a line is The solution, in the case where the needle length is not greater than the width of the strips, can be used to approximate .
• Ham sandwich theorem in mathematical measure theory states that for every positive integer n, given n measurable "objects" in n-dimensional Euclidean space, it is possible to divide all of them in half (with respect to their measure, i.e. volume) with a single (n − 1)-dimensional hyperplane. The theorem takes its name from the case when n = 3 and the three objects of any shape are a chunk of ham and two chunks of bread—notionally, a sandwich—which can then all be simultaneously bisected with a single cut (i.e., a plane). In two dimensions, the theorem is known as the pancake theorem because of having to cut two infinitesimally thin pancakes on a plate each in half with a single cut (i.e., a straight line).
• Fermat's Christmas theorem in additive number theory states that any odd prime p is expressible as p = r ² + s ², where r and s are integers, if and only if the remainder on dividing p by 4 is 1, written p = 1 (mod 4). For example, the primes 5, 13 and 17 are congruent to 1 modulo 4. They can be expressed as sums of two squares: 5 = 1² + 2², 13 = 2² + 3², 17 = 1² + 4². Fermat wrote an elaborate version of the statement (in which he also gave the number of possible expressions of the powers of p as a sum of two squares) in a letter to Marin Mersenne dated December 25, 1640. For this reason, this version of the theorem is sometimes called Fermat's Christmas theorem.
• 7 is the only prime followed by a cube. Here's the proof: x³-1=(x-1)(x²+x+1) and this number is composite unless x-1=1 ⇒ x=2 and x³=8
• The Fields Medal was designed in 1933. In the front of the medal, we can see a portrait of Archimedes a quote and the date in Roman numerals. If you look closer you will notice that there seems to be a typo in the date: MCNXXXIII instead of MCMXXXIII.
• An easy to remember approximation for the number of seconds in 1 year is .
• What if I say that the next number in the sequence 2,3,4 is 82000? It’s true if we define the sequence such that the nth integer in the sequence is the smallest number > 1 whose representation in all bases up to n consists only of zeros and ones.
2 in base-2 is 10
3 in base-2 is 11, in base-3 is 10
4 in base-2 is 100, in base-3 is 11, in base-4 is 10
82000 in base-2 is 10100000001010000, in base-3 is 11011111001, in base-4 is 110001100, in base-5 is 10111000
• 1, 2, 4, 8, 16, _? What if I say the answer is 31 instead of 32? It’s true if we define the nth term of the sequence as the number of pieces a circle is divided into by lines connecting n number of points on the circumference of the circle.
I think that’s all! :)
Image source : Google, Wikipedia, Facebook, Pinteres
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