Everybody is introduced to mathematics by “remembering” natural numbers. Today we will see, “How to construct natural numbers”.
Without much delay let’s construct “Natural Numbers”, as in “plane geometry” (you must have studied it in high school), we define “Euclid’s Axioms”, here we will define “Peano’s Axioms”, the axioms are:
- Axiom One: 1 is a natural number.
- Axiom Two: Every natural number has a successor.
- Axiom Three: 1 is not the successor of any natural number.
- Axiom Four: If the successor of equals the successor of , then equals .
- Axiom Five (Principle of mathematical induction): If a statement is true of 1, and if the truth of that statement for a number implies its truth for the successor of that number, then the statement is true for every natural number.
Some texts, which swap meaning of natural numbers and whole numbers, consider natural numbers to include the number zero! In that case, we will start our construction with zero instead of one (see: http://mathworld.wolfram.com/PeanosAxioms.html). This supports a very criticized philosophical idea from Indian civilization:
Everything started from nothing.
Once we are able to construct natural numbers, we can construct all kinds of number systems (with motivation of solving certain algebraic equations):
- whole numbers: add ‘0’ to the list of natural numbers (I don’t know why, but some mathematicians use whole numbers and natural numbers interchangeable)
- integers: add negative natural numbers to the list of whole numbers
- rational numbers: make fractions from integers, where denominator is not allowed to be zero.
- irrational numbers: not all numbers can be represented as fractions
- real numbers: define “Dedekind cut”, and construct real numbers out of rational numbers (lengthy task!)
- complex numbers: any polynomial equation must have a solution
There is another classification of “real numbers” , called “algebraic numbers” and “transcendental numbers“, but it is altogether a different topic of discussion which I will discuss in some other blog post.
Pascal’s triangle has long and celebrated history, see this TedEd video:
What makes it more interesting is its relations with various domains of Mathematics (if you don’t understand the three relations discussed, refer Wikipedia). Here I will point few such connections:
1. Fibonacci Numbers: I believe that this is the most celebrated observation from pascal’s triangle. To see the jungle of Equations involved, visit http://www.maplesoft.com/applications/view.aspx?SID=3617&view=html
2. Chebyshev Polynomial: We can find coefficients of Chebyshev Polynomial using pascal’s triangle. See- http://mathpages.com/home/kmath304.htm
3. Ramsey Number: Upper bound of Ramsey Number can be found using Pascal’s triangle, for more details refer : https://plus.maths.org/content/friends-and-strangers
Why a layman should care about RSA: Your all “secure” online transactions are based on RSA encryption!
RSA stands for the initial letters of the surnames of Ron Rivest, Adi Shamir, and Leonard Adleman, who first publicly described the algorithm of one of the first practical public-key cryptosystems which is widely used for secure data transmission. In this blog post I will try to illustrate the RSA algorithm using a puzzle (that I came across during my summer internship in Delhi):
Suppose there are two scientists, M and F (assume M to be male and F to be female for ease of writing), collaborating on a classified research. Both live far apart and F wants to send a chemical for testing to M. She designs a box with two latches for locking (see photo below), so that two different locks can be used to lock the box. [Assume that if any lock is tried to be broken, the chemical will spill and evaporate].
Two such latches are there
Now, F locks one of the latches and keeps the key with herself and sends the box to M. When M receives the box, he locks the other latch and keeps that key with himself.
M then returns the box to F. When F receives the box, she unlocks her lock and sends the box again to M. When M again receives the box, he unlocks his lock.
Voilà! the chemical has been securely sent by F to M and that too without exchanging the keys.
If we make the above physical “lock” an abstract entity (i.e. numbers!), and minimize the possibility of breaking lock (like in above illustration, breaking lock caused loss of chemical), what we get is the RSA encryption.
For Mathematics behind RSA encryption refer: http://blogs.ams.org/mathgradblog/2014/03/30/rsa/#sthash.ce5YIAO6.dpbs
Cover page of my report.
As promised in my earlier blog post: “Conclusion (non-mathematical)” , here is the link for my project report: https://gaurish4math.wordpress.com/downloads/technical-report/
If interested post your review as comment below