We don’t think too much about numbers, as long as they show upward moving trend in our bank accounts, isn’t it? We have used numbers to add, subtract, multiply and divide. Most of us have used numbers as a means to learn and understand bigger things. We have taken numbers for granted!
However, there are mathematicians, or shall I call them number mavericks, who have devoted their entire life and professional career to understanding the numbers - the symmetry, the patterns, the relationships between them. The number mavericks aren’t the result of any special training, they are born with the ability to see the patterns where none appears to others. They can keep more in their heads and process it, without having to put anything on paper, than majority of us. They simply have more evolved left brain functions! They love the numbers and I guess it is retuned in kind by the numbers.
For this article, I have picked up three concepts related to numbers that I found interesting - Golden ratio, Perfect Numbers and Prime numbers.
Golden Ratio
Have you ever looked at a piece of art or a building and have been mesmerized by how proportionate everything looks? Well, most likely the proportions are in golden ratio.
Two quantities, a and b, where a > b, are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. This is also called “Phi”.
(a + b) / a = a / b = Phi (
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The golden ratio is approximately 1.6180339887….
Artists are known to use golden ration to define the proportions of their work. For example, defining the height of a building façade with respect to its breadth. If the building breadth is 100 meters, the height of 161.8… meters would make the building proportionate as per golden ratio and pleasing to the eye.
UN headquarters building in
There are a number of cool properties of this number. To cite a few, to obtain reciprocal of phi, subtract 1 from Phi, and you get 0.6180339… which is 1/Phi. To get square of the golden ratio, add 1 to Phi, and you get 2.6180339…, which is (Phi)2
The phi appears everywhere in nature. The proportion of human body, of many other animals, plants and even the Solar system seems to be governed by the golden ratio.
Two number series progressively provide numbers that generate increasingly accurate golden ratio. These number series are - Fibonacci series and Lucas series.
Fibonacci numbers – 0,1,1,2,3,5,8,13,21,34,55,89,144,233…… The series starts with 0,1 with each next digit as the sum of previous two.
Lucas numbers – 2,1,3,4,7,11,18,29,47,76,123,199…… The series starts with 2,1 with each next digit as the sum of previous two.
The importance of these series is the fact that as you go further down these series, the furthest two quantities that you pick will increasingly be closer and closer to the golden ratio.
Perfect Numbers
Numbers whose divisors, excluding the number itself, add up exactly to the number itself are perfect numbers. e.g. 6 is a perfect number, its divisors 1, 2 and 3 add up to 6. The next perfect number is 28.
Perfect numbers are always the sum of a series of consecutive counting numbers
6 = 1+2+3
28 = 1+2+3+4+5+6+7
496 = 1+2+3+…..+30+31
Perfection of the perfect numbers is closely linked to the “twoness”, all the numbers that are power of 2. 2n only just fail to be perfect, because sum of their divisors always adds up to one less than the number itself. This makes them slightly defective
22 = 4; Divisors 1,2; Sum = 3
23=8; Divisors 1,2,4; Sum = 7
24=16 Divisors 1,2,4,8 Sum = 15
6= 21 x (22 -1)
28 = 22 x (23 -1)
496 = 24 x (25-1)
Although there are plenty of numbers that are slightly defective, that is whose divisors add up to 1 less than the number itself, however, there appear to be no number which is slightly excessive!
An interesting anecdote about the perfectness of the perfect numbers - It is said that God created the world in 6 days. The perfectness of 6 is not derived from that fact that God chose 6 days to create the world, on the contrary, since 6 is perfect; God took 6 days to create the world!
Primes Numbers
Prime number is a natural number that has exactly two divisors, as the mathematicians say, trivial divisors, 1 and itself. No other number on the number line can divide a prime perfectly.
Prime numbers are mother of all positive integers greater than 1. All positive integers can be created by multiplying a unique sequence of primes. This is called unique factorization.
All prime numbers can be put in 2 categories (except 2). One category that fits 4n+1 and 2nd category that fits 4n-1. First category of primes are always the sum of two squares e.g 13 = 22 + 32 . The 2nd category primes can never be written as sum of squares of two other numbers.
Another interesting property of prime numbers is the random fashion in which they appear on the number line. As you travel along the number line, it is not possible to predict if the next number will be a prime or not. Mathematicians have spent innumerable number of years trying to predict the pattern, but in vain.
As you go further down the number line, and as the numbers become big, it becomes increasingly difficult to find prime factors, to prove whether the number is a prime or not.
There are a number of popular notions around prime numbers. These numbers are supposed to have in them the code from the super beings. Once we are able to predict the pattern, we would have also predicted the unseen and other mysteries that seem beyond logic.
It is also believed that any other intelligence species inhabiting any other part of the universe would also have figured out the uniqueness of primes, and hence, if they try to communicate outside their world, they will send the messages encoded in primes, the same that we humans do. Our signals that go out to look for other intelligent species are encoded in primes.
One very practical and increasingly relied upon use of prime is in cryptography. The most widely used cryptography algorithm uses prime numbers. The algorithm has a private key and a public key. Here’s how it uses primes.
Two very large prime numbers are multiplied to get a third number. The pair of prime numbers is called the “private key” and the third number created as a result of multiplying them is called the “public key”.
Public key is used to encrypt. However, for decrypting, the private key, the two prime numbers, that were originally used to create the pubic key, is required.
The strength of this algorithm depends on the fact that though it is possible to factor the public key to arrive at the private key, but with sufficiently large prime numbers as the private key, it is impractically. In other words, the strength of the algorithm is based on the fact that it is easy to multiply two large prime numbers, but very difficult to find its prime factors. Mind you, most of the web security infrastructure relies on this fact.
Having said that, the race is on for a clever solution that can make it easier to find factors. Once that happens, make sure you stop transacting on the web, till the web security infrastructure is re-hauled!
Here’s your chance of making some money – Find a really big prime, a prime that has at least 10 million digits!! Some research foundations have instituted huge prize for finding larger and larger primes. Here’s the hint – You can go about this hunt in either of the two ways – One is “brute force” - Pick 10 million digit numbers, one after another, of course you will use some basic maths to eliminate the obvious composites (All natural numbers that are non-primes are composites) and start finding factors till you hit a number that has no factors and that is your prize winning prime! 2nd method is to come up with a clever way to find factors. Apply that method to 10 million digit number and find your prize winning prime!
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Happy hunting!
