The Siren Call Of Mathematics
(Or How To Be As Cheerful As The Girl Who Has Inside Scoop on Gossip!)
Where From Here?
The first love you never got around to (fully) express tugs at your heart like no other thing! For me, it was mathematics.
Mostly life does not let you love with abandon.😀 At the very least, you got to do a lot of other things so that you may live, and then possibly have a chance at love.As this year comes to an end, I walk down that lovely promenade of mathematics memories.
Fortunate to have read these books(long back), absolutely wonderful!
Just reading them had kept me happy for many days, feeling intellectually pompous and as cheerful as the girl who has the inside scoop on the gossip!
The effect is most strange, given that I am not certain if I am going to come up with an original mathematical proof myself or even if I am understanding the proofs completely or exactly as the mathematicians intended them to be understood.But I always felt like a member of that community, as if I was the Fermat from the past.So maybe..😄
Curiouser and Curiouser: The Curious Numbers!
Go down deep enough into anything and you will find mathematics.Go down deep enough into mathematics, and you will find numbers!
The best things about numbers is that they are the foundation of all things that follow, they let us in a mystery, letting us see the inside scoop of all the things!
Natural numbers, even numbers, odd numbers, whole numbers, rational numbers, real numbers, imaginary numbers, complex numbers, the list goes on..
Numbers, numbers, numbers: What Are They Upto?
Prime number: Numbers that are not divisible by anything other than 1 and itself. Whether its cryptography or simple hashing, odd primes are most valuable!
Twin Prime numbers: When the subsequent primes are separated only by a single number, they are called twin primes. For eg: (3, 5), (5, 7), (11, 13) are twin primes.An isolated prime is a prime number p such that neither p − 2 nor p + 2 is prime. For eg: 2, 23, 37, 47, 53, are isolated primes.
Composite number: Definitely not a prime! If a number has three or more divisors, its a composite number.
Fun Fact: There are even and odd prime numbers.But prime number 2 is the oddest prime number as its the only even number to be a prime! 😃
Perfect number: A number which is also equal to sum of all its proper factors.For eg, 6(1+2+3 = 6) All perfect numbers are even.
Proof Pending: Is there “ever” a odd perfect number?
Fun Fact: A semi perfect number is number which is sum of “all or some” of its proper factors. Similarly there are abundant and deficient numbers.Abundant numbers are one whose sum of factors exceeds the number itself.For eg, 12(1+2+3+4+6 =16 > 12). Deficient numbers are those whose sum of factors is always less than the number itself. For eg, 21 (1+3+7 =11 < 21).A natural number that is abundant but not semi perfect is called..Yes, you guessed correctly..a weird number. For eg, smallest weird number, 70(1+ 2+ 5+ 7+ 10+ 14 + 35 = 74, but no subset of these factors sums to 70, hence abundant but not semi perfect, weird!!)
√2: First known irrational number, or whose representation is not known precisely.~1.4142135623730950488016887242096980785696718753769480731
Fun Fact: There is no rational or precise way to express side of right triangle whose other two sides are 1 unit each or for the diagonal of square whose each side is 1 unit each. At the time, Pythagoras and the nerd squad were still trying to come to terms with irrational numbers.If the digit that comes next could not be known for sure, maybe then even God does not know it. This seemed in addition to be mathematically confounding, outrageous to their philosophy and religion! Hippasus, the mathematician who tried to expose this “secret” to public was killed. In the name of math, there has been bloodshed.
Imaginary number: If we imagine the square root of a negative number, we have imagined an imaginary number 😃 square(i) = -1
Fun Fact: That its imaginary 😃
π: Most ubiquitous, most famous! It is ratio of circumference over diameter of a circle. An Irrational number~3.1415929.
Proof Pending: The sequence of digits in Pi have statistical randomness.They definitely “seem” to have randomness, but not proven.
Fun Fact: March 14(3/14) is celebrated all over the world as the “Pi Day”. People are still so fascinated by π , they compete for world record to recall π decimals Memorize million decimals of π and beat that record! 😃
Triangular numbers: The sequence of numbers that can be represented in a triangle.The nth triangular number is n(n+1)/2.
Fun fact: When mathematician Carl Gauss was very young, he was asked to sum 1 to 100 numbers, he immediately created 50 pairs in his mind and summed using this property. 1+2+3+4+..+n = n(n+1)/2 = 50 * 101 = 5050
Pascal Triangle: The Pascal triangle is formed of numbers by adding two consecutive numbers from preceding row.
Fun Fact: Though it is attributed officially to French mathematician, Blaise Pascal, it had been in use since 10th century! Pascal formalized it when helping someone on a gambling dice problem.It has several patterns inside it, forms the basis of combinations and probabilities.
Fibonacci numbers: F(n) = F(n-1) + F(n-2) with F(0) = 0 and F(1) = 1 (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89,144,..)
Proof Pending: Are there “infinitely many prime” Fibonacci numbers?
Fun Fact: The sum of shallow diagonals in Pascal triangle yields the fibonacci numbers! Every fourth fibonacci number is divisible by 3, also the only multiples of 3 are every fourth fibonacci number (3, 21, 144..)
Golden Ratio number: Φ (~1.618). Φ is the only number which has the mathematical property of its square being one more than itself. 1.618… + 1 = 2.618…
Fun Fact: Fibonacci numbers divided by its predecessor converges to golden ratio! Golden ratio is seen in many architectures.
Magic squares: The magic squares are n*n square grid filled with distinct positive integers such that each cell contains a different integer and the sum of the integers in each row, column and diagonal is equal.
Fun Fact: Its worth a mention, as we have been preoccupied with it since centuries.Sudoku, Rubik’s cube! The reason is that once solved, it represents perfect distributions in many ways and perfect harmony of numbers.
Hailstone numbers: Take any positive integer n. If n is even, divide it by 2 to get n / 2. If n is odd, multiply it by 3 and add 1 to obtain 3n + 1.
Proof Pending: That no matter what number you start with, you will always eventually reach 1.
Fun Fact: Because there is a distinct ascent and descent in the sequence of numbers thus generated, these numbers are called hailstone numbers!😃
Bernoulli numbers: The numbers showcasing recurrence pattern in sum of squares, sum of cubes, etc are Bernoulli numbers. The generalized form of Bernoulli number is defined as -
Fun Fact: Bernoulli numbers (formulated by Jacob Bernoulli ) is different from Bernoulli principle of Hydrodynamics (stated by Daniel Bernoulli ).To add to the Bernoulli confusion, they both are mathematicians of Swiss origin!
Catalan numbers: The sequence of numbers formulated when thinking of finding ways of dividing a polygon into triangles.
Fun Fact: In a party of n, number of people shaking hands simultaneously without crossing arms leads to Catalan numbers !
Kevin Bacon numbers: The number of degree of separation from Kevin Bacon, using shortest path algorithm.
Fun Fact: Kevin Bacon said he had worked with everyone in the industry, and so started to be called as center of universe.Perfect recipe for fun.
Carmichael numbers: They are composite numbers having prime factors and square free.For every prime factor p of n, p-1 exactly divides n-1. For eg, 561 is the first Carmichael number.
Fun Fact: Carmichael numbers are important because they pass the Fermat primality test but are not actually prime!
Hardy–Ramanujan number: 1729 is Hardy-Ramanujan number, the smallest number that can be expressed as sum of two different cubes.
Fun Fact: This Hardy-Ramanajum story confirms that every number has some interesting tale to tell! It is also the third Carmichael number.
What’s more there are so many more numbers, amicable numbers, sociable numbers, strange numbers, transcendental numbers, perfect squares, obstinate numbers, and the list goes on..Every number has a tale and is interesting.If it were not, that would make it interesting!
Hope you enjoyed this walk! In the next of my mathematics series, we will walk down the Euler Walk, sieve the Sieve of Eratosthenes, find ourselves confounded by Russell’s Paradox, ask the twenty questions off a tree, and ponder why the procedure to bring Rubik’s cube from any random position to its solved state in the minimum number of steps is the ultimate God’s algorithm!
All said, this article is more for me than for you.
But if it reminds you of what you loved and forgot all about, well then it is for you too! 😀
The year is ending. Naturally you rush to get just one more thing done before its all done! Thus came out my first medium post the last day of the year! 😊
Wish you all a very Happy New Year! 😀
