Maths... it is just a wonderful subject. Highly addictive and informative, maths is. Mathematics defines the universe as we know it in a more sensible way than words. Words generally resort to abstract notions and fancies while describing the universe, but maths... Maths is more concrete. It has structure, it has substance, it makes perfect sense. In the universe, maths signifies order and foundation. The numbers, they never lie. It has no grey zone, it is either this or that. It does not leave you guessing too much. That is why numbers have completely different feel than words.
I was compelled to write about maths because of a joke my friend told me. It goes as follows: Which is the odd man out among the given terms: a mathematician, an electrical engineer, a plumber and a 12-inch pizza?
Answer: The mathematician, because he is the only one who can't feed a family of four...
This joke, this stupid joke is so bad... that I found it funny only later, after I had ranted sufficiently about the greatness and beauty of maths.
I am good at maths, no doubt, but I would really like to know more about numbers and the research people are doing just to beautify the way we see numbers and the number system. That is, I would love to learn Pure Mathematics, not just the applied stuff. I have been studying a lot of Number Theory lately and I have come to know of operations and functions which have very little or no use in physics or chemistry, but are powerful tools in solving sums based on pure mathematical research. The simple stuff we learn in school like prime numbers, at that time we find them to be just a bunch of numbers which have no natural divisor other than 1 and itself, but if you go deeper into that concept of prime numbers and start finding bigger and bigger prime numbers, it all seems wondrous! There is no pattern for the occurrence of prime numbers nor is there a formula or algorithm to find prime numbers, but that is what makes it all the more interesting! There is no pattern, there is utter chaos! And that is why finding order is of more use and of more fun! There are an infinite number of prime numbers (yes, that has been proven), so people are in constant hunt for newer and larger prime numbers. The largest prime number has some 1,74,25,170 (I use the Indian system of commas) digits! This number was found using the Mersenne method of finding primes, that is, in simple terms, a number expressed in the form of: 2n-1. If it ends up as prime, great! If not, try another number - is the idea in layman's terms.
I have taken a great interest in finding out new ways to factorize numbers and I have found a way to factorize large numbers with large prime factors quite efficiently. Only later did I realize, that this was one of Fermat's many algorithms... but anyway, I found it on my own, although not new to the world, but to me, so I am proud of myself! I state this as (this is some technical stuff, you can skip this if you want, for you could say I am just showing-off here):
I was compelled to write about maths because of a joke my friend told me. It goes as follows: Which is the odd man out among the given terms: a mathematician, an electrical engineer, a plumber and a 12-inch pizza?
Answer: The mathematician, because he is the only one who can't feed a family of four...
This joke, this stupid joke is so bad... that I found it funny only later, after I had ranted sufficiently about the greatness and beauty of maths.
I am good at maths, no doubt, but I would really like to know more about numbers and the research people are doing just to beautify the way we see numbers and the number system. That is, I would love to learn Pure Mathematics, not just the applied stuff. I have been studying a lot of Number Theory lately and I have come to know of operations and functions which have very little or no use in physics or chemistry, but are powerful tools in solving sums based on pure mathematical research. The simple stuff we learn in school like prime numbers, at that time we find them to be just a bunch of numbers which have no natural divisor other than 1 and itself, but if you go deeper into that concept of prime numbers and start finding bigger and bigger prime numbers, it all seems wondrous! There is no pattern for the occurrence of prime numbers nor is there a formula or algorithm to find prime numbers, but that is what makes it all the more interesting! There is no pattern, there is utter chaos! And that is why finding order is of more use and of more fun! There are an infinite number of prime numbers (yes, that has been proven), so people are in constant hunt for newer and larger prime numbers. The largest prime number has some 1,74,25,170 (I use the Indian system of commas) digits! This number was found using the Mersenne method of finding primes, that is, in simple terms, a number expressed in the form of: 2n-1. If it ends up as prime, great! If not, try another number - is the idea in layman's terms.
I have taken a great interest in finding out new ways to factorize numbers and I have found a way to factorize large numbers with large prime factors quite efficiently. Only later did I realize, that this was one of Fermat's many algorithms... but anyway, I found it on my own, although not new to the world, but to me, so I am proud of myself! I state this as (this is some technical stuff, you can skip this if you want, for you could say I am just showing-off here):
Let x be the number you have to factorize. If x is even, then obviously, 2 is a factor and that simplifies your task a bit. But if x is odd (you might face the same problem once you have divided x by 2 and received a weird odd number), and is composite mind you, it will have two odd numbers as it's factors. Let those odd numbers be p and q. Those two numbers will have a natural Arithmetic Mean. Let that be denoted by a. Now the arithmetic mean will differ from p and q equally i.e. p = a + b, q = a - b. (Property of arithmetic mean). So let b be that constant difference. Therefore, x = pq ⇒ x = (a+b)(a-b) ⇒ x= a2-b2 ⇒ a2-x=b2
So any composite number can be denoted as a different between two squares and can be factorized. But to find the number from whose square the number has to be subtracted requires a little trial and error. You take the square root of the x and take the smallest natural number greater than the square root and subtract x from it and check whether the obtained number is a perfect square using a simple calculator. This method does not take long if you use a calculator and you don't have to try too many values, the answer comes in approximately 6 tries.
Yes, such tedious methods too time-consuming and a number can be factorized just by using an app on a smartphone or on the net. But I just want to ask, what is the fun in that!? If you are given a problem and the other guy knows the solution and is ready to tell it to you, would you just listen to the solution without even trying!? If you would, you are an idiot, seriously. That is no way treat a problem or your mind. If you don't stress your head, it will degenerate into a dead weight. Don't let that happen.
Coming back to maths, such stuff about factorizing numbers is really fascinating and fun thing to do. You start seeing new connections between numbers and you will find working with them easier. For a maths-loving guy like me, I do this a lot of times. Sitting in the back of the car, having nothing to do, I just look at the number plates of other cars and start factorizing the number! I have proved lots of tests of divisibility by doing this!
Another beautiful concept of maths is of infinity. The end of the number line. The place where parallel lines meet... So basically the point where the world is damned and where regular rules of algebra do not apply. How beautiful is that! What I really find interesting are these infinite series... Simply magnificent, they are! Here is the thing with infinite series', they are infinite and so cannot be perfectly rationalized. You can't seem to use the answer of "infinity" for it is too weird. So what we humans have done is, instead of using infinity itself, we have decided to manipulate it. Play with it a little bit, use its paradoxes against itself and come up with logical answers for infinite series. For example, the sum of all natural numbers from 1 to the natural infinity is not infinity. It is in fact -1/12! What the fuck! I know, right! Ahh... it is brilliant. It defies logic, really. But the methods are logical... So what do we do now? I really don't know, yet...
Maths has some really amazing concepts, like Pi for example. Pi is the ratio of the circumference of a circle to its diameter. Simple enough, but the ratio is not rational. Fucked up, right? We define rational numbers as numbers which can be expressed as a ratio of two numbers i.e. m/n where n is a non-zero number. But pi, that bitch, is an irrational ratio. How wonderful! It has no definite value, per se. But for use in practical life, we can use approximation, they do work.
But from a mathematical point of view, approximations don't work. It needs to be concrete and precise. You can give me a value of pi up to a gazillion digits, but it is not going to work because there will be some stuff after the gazillionth digit, too. It goes on and on and on...
Approximations don't work for maths. Patterns work for maths, yes. If you manage to find a pattern in the digits of pi, then too, it will be a great leap in mathematics. We will never have a definite value of pi, but lets go closer and closer to it just to enhance the beauty of the subject. That is the main aim of Pure Mathematics... Money does matter, but... the universe is richer than money itself... For maths, just seeing stuff is not enough. You have to go deeper. Relatively simple concepts like prime numbers have such great depth and importance to it! We just have scratched the surface. To know more about the universe, just a glimpse or glance is not enough. Everyone sees, but only a few observe...
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