Showing posts with the label Mathematics

Book Review: Beyond Infinity

Geeking out is something I yearn to do from time to time. Having seen a documentary on infinity from Netflix last year, I decided to pick up a book on the topic.  Beyond Infinity , by  Eugenia Cheng , is an excellent guide delving into this mind-boggling notion. In this book, Cheng explores some key aspects of infinity that challenge our intuition and understanding of mathematics: Numbers Numbers Numbers, natural, rational, irrational and real:   Cheng explains that these number sets are infinite in nature but some are "more infinite" than others.  For instance, the set of natural numbers are smaller than its superset of rational numbers, which in turn are smaller than the set of real numbers.  So far so good?  But...  Infinity is but an abstract notion: It certainly is not a number   to which the rules of arithmetics apply: ∞ + 1 = ∞ (addition/subtraction does not apply) 2  ⋅  ∞ = ∞ (neither does multiplication/division) 1/∞ = 0 (in the sense of limits which touches on the f

A Very Real Life Application of the Stable Marriage Problem

I recently stumbled upon a mathematical problem known as the  Stable Marriage Problem (SMP).  Per Wikipedia, the problem is commonly stated as: Given N men and N women, where each person has ranked all members of the opposite sex with a unique number between 1 and n in order of preference, marry the men and women together such that there are no two people of opposite sex who would both rather have each other than their current partners. If there are no such people, all the marriages are "stable". The SMP has real-life applications to any problem requiring stable pairing of two sets of equal size. In fact, this problem is always solvable using the Gale-Shapley algorithm . There is a rather big catch however.  While the marriages are always stable, they may not be ideal from the vantage point of an individual.  To illustrate this, imagine three men A,B,C and three women X,Y,Z. Here are their ranked preferences for members of the other group: A: YXZ B: ZYX C: XZY X

9 Equations a True Geek Should Know

Came across this  article  which is entertaining to read while keeping score!  I scored 2 of 9 - Uncertainty Principle and Maxwell's Equations. The good news is I do not qualify as a geek, but that also means there is so much more to learn when I retire. -PTS

Prime Numbers and Encryption

Want to make   $250 000 ?  Find a big prime number, a really big one. It turns out there are organizations  ready to dough out good cash for a really large prime number.  This is because primes are used in   RSA cryptography . RSA Algorithm Let's look at the algorithm: 1.       Multiply two large prime numbers  p  and  q  to get the product  N 2.       Find two numbers  e  and  d , such that  ed = 1mod((p-1)(q-1)) , where  e  and  N  are relatively prime meaning they do not share any prime factors. 3.       Let's call  M  the original message and  C  the ciphered message:  a.       To encrypt: C = M e mod(N) b.       To decipher: M = C d mod(N)  In essence, using the public key   ( N,e)  will transform the original message   M   to the ciphered message  C . On the contrary, applying the private key  d  on the ciphered message   C  will result in the original message   M .  Security of Encryption The beauty of RSA is your public key can be published for anyone to