Numbers are truly fascinating concepts and arguably one of the most important discoveries which have helped mankind to progress and achieve the prosperity we are cherishing as of right now; even when you are reading this sentence. Today, numbers are so deeply rooted in our society that students are compelled to memorize them right from their childhood which makes them believe in a false notion that numbers are physical entities (like a leaf, or a stone) which can be measured and sensed. One has to understand that numbers are really just concepts and abstractions. What makes the concept of numbers unique is the fact that each number is distinct and it can not be, any other number except for itself.
As the theory of numbers developed, mathematicians discovered that some numbers have peculiar properties and today we are going to look at 5 such numbers which possesses intriguing properties:
- 0 (Zero)
Zero lies in the classification of Integers (Z) and (N). This number had been perplexing mathematicians since a very long time ago because it would invoke the age old philosophical question “How can nothing be something?”. The Romans did not include the number zero in their numeral systems at all however, it was later introduced in the Hindu-Arabic system and people have been accepting its existence ever since. Furthermore, Zero demonstrates rather weird behavior when it is treated with arithmetic operators such as :
N + 0 = N | 0! = 1 | 0÷0 = undefined | 0^0 = undefined | N x 0 = 0
It would seem as if the concept of numbers is nullified by 0 which makes it included in our list.
- i (iota)
Here is a number that has been boggling the minds of high school students since its inception. Perhaps the word “Imaginary number” contributes to the fact that people think this is a number which transcends from the categories of numbers and somehow is irrelevant and unnecessary to understand. This could not be more further from the truth. Iota “exists” just as much as other numbers like 1, 25 or 4.7 do. They are all just that, pure concepts formed in our minds. It just so happens that we are more comfortable with integers and fractions and use them on a daily basis and as soon as we are introduced a novel concept we tend to reject it.Regardless of the opinions of people who don’t appreciate its essence, iota is associated with some strange properties as well. iota is defined as square root of negative 1.
i = √-1 .Then the question arises is what is the square root of i itself. In mathematical terms ,
√i = ? .
Here, is the answer to the question with its derivation. Furthermore, it seems astounding that reciprocal of i is equal to negative i. Mathematically,1 ÷ i = -i (Why not think this one through yourself ?)
Initially thought to be useless and inapplicable in the real world, today this is number is vital in modelling graphs and understanding the behaviors of waves.
- e (euler’s number)
If you were patient enough to sit through the entire class of calculus, you will have surely been introduced to this number. And rightfully so, you should because, more interesting than the number e, is its function ex
The useful property of this graph is that the value of the function (height of green line) will always equal to the slope of the graph at that particular point where it intersects in the function. Furthermore, it also equals to the sum of all the areas of squares you see below the red line. You can represent the sentence above in an equation as :
∫ exdx = ex + c and d(ex) / dx = ex
Euler’s number itself can be defined in various ways such as:
or in terms of infinite sequences as :
- π (Pi)
Pi probably is the most famous irrational number and it would not be an understatement to say that pi appears everywhere. Definition of pi is that it is the ratio of Circumference with respect to the diameter of a circle. Although pi originates from circle, the fact that it appears in almost all sectors of mathematics and physics is simply astounding. Some of the few areas where it is absolutely required are, probabilistic models, harmonic motions, number theory, calculus and also the infamous euler’s identity,
Being an irrational number the digits of Pi are random and infinite. However, this does not stop mathematicians and computer scientists alike from computing the digits of pi.There are even competitions held on March 14 (3/14) known as Pi day where people memorize the digits of the number and some of them have been successful in reciting more than 70,000 digits of the number pi without error.Just like e, Pi also can be represented as sum of infinite series :
and many more.
- φ (Golden ratio)
The golden ratio is denoted by the Greek alphabet phi and it is intimately associated with the Fibonacci sequence. Numerically the value of golden ratio is :
But its frequent appearance in nature, geometry, music and even arts has baffled the minds of mathematicians. Golden ratio is needed to create geometrically uniform patterns and therefore it is believed that shapes which contains the golden ratio are aesthetically pleasing to the eyes.
Golden ratio is overwhelmingly present in nature as well. From sunflower’s spiral to sea shells and even including the very galaxy that we are living in.
We might never be able to decipher the secrets of all the numbers there ever are, still we can always ponder upon the origin and meaning of such numbers and strive for perfection just as the numbers do. Philosh for example, tries to solve problems with solutions that are simple, elegant and aesthetically pleasing; just like φ is.
Do you agree with our list? What is your favorite number and what is secret behind its origin?