Most of the times I have blogged whenever something has bugged me. I have railed against - Internet service providers, bankers (here, here and here), statisticians, entire Countries (Greece and India) , analysts, global warming, chartered accountants, Tambrahms etc.

Just to balance things out a little bit, I am going to write about a few things that have amused me. I am going to focus on mathematical ideas that I have seen recently which have held my attention. Apparently, the Pythaogoren brotherhood used to go around behaving like a "cult", looking for mathematical patterns everywhere. I am going to list a set of popular references and interesting math bits here.

My favourite in this whole lot are the "almost perfect" numbers. They, um, remind me of myself. And yeah, if you did not know before, number geeks are extremely fond of powers of two.

There is only one natural number in the world, whose successor is a cube and predecessor is a square. Finding this number is not that tough. Proving it is fiendishly tough (Apparently. How would I know?). Fermat apparently mentioned about this number in some letter that he wrote.

There is only one 4-digit perfect square that is of the form 'aabb' where a and b are digits from 0 to 9.

Some other interesting nuggets - there is only one set of three consecutive odd integers all prime (Find these). We can find a set of 6 integers in AP all of which are less than 1000 and are prime (Find these as well, this is tougher). 16 is the only natural number that can be represented as x^y and y^x, where x, y are distinct. Who woulda thunk?

An irrational number raised to an irrational power can be rational (unlike probably a lot that I have mentioned in this post). Try proving that.

Every natural number in the world has a multiple that comprises all the digits appearing at least once each.

Did you know that two triangles can have 3 angles equal and 2 sides equal and still be not-congruent to each other? A quadrilateral can have a pair of sides equal and a pair of sides parallel and still not be a parallelogram. I think it is easier to be this quadrilateral than to be those two triangles.

Do you know that we put our kids through 15 years of school education without them discovering most of these facts? Let me stop right there. I can sense a rant against the Education system coming through the system.

Just to balance things out a little bit, I am going to write about a few things that have amused me. I am going to focus on mathematical ideas that I have seen recently which have held my attention. Apparently, the Pythaogoren brotherhood used to go around behaving like a "cult", looking for mathematical patterns everywhere. I am going to list a set of popular references and interesting math bits here.

**Ramanujan Number**: Many might have heard about this. This is the number 1729. It is special because this is the smallest natural number that can be split as sum of two cubes in two different ways. 1729 = 12^3 + 1^3 and 10^3 + 9^3. It must take a particularly brilliant mind to stumble upon this. Now, what is the smallest number that can be broken as the sum of two squares in two different ways? What is the smallest number that can be broken as the sum of two squares in two different ways if the squares have to be distinct?

**Armstrong Number:**A 3-digit Armstrong number is a number that is the sum of the cubes of its digits. 153 is an Armtrong number. 1^3 + 5^3 + 3^3 = 153. There are a few more. Life is short. It will never feel complete if you do not know all the Armstrong numbers. Give it a go.

**Perfect Numbers:**, A perfect number is one that is equal to the sum of its factors (except itself of course). The talk on perfect numbers takes us on to numbers that are semiperfect, deficient, abundant or amicable. Some which are abundant but not semiperfect are called weird, as one can clearly see.

My favourite in this whole lot are the "almost perfect" numbers. They, um, remind me of myself. And yeah, if you did not know before, number geeks are extremely fond of powers of two.

**The best number in the world**is 73. This is from Big Bang Theory by the way. 73 in binary is a palindrome. 73 is the 21st prime, which by itself is not such a big deal, but if you reverse 73, it gives us 37 which is the 12th prime.The digits of 73, 7 * 3 gives us 21, which is why being the 21st prime is such a neat deal.

There is only one natural number in the world, whose successor is a cube and predecessor is a square. Finding this number is not that tough. Proving it is fiendishly tough (Apparently. How would I know?). Fermat apparently mentioned about this number in some letter that he wrote.

There is only one 4-digit perfect square that is of the form 'aabb' where a and b are digits from 0 to 9.

Some other interesting nuggets - there is only one set of three consecutive odd integers all prime (Find these). We can find a set of 6 integers in AP all of which are less than 1000 and are prime (Find these as well, this is tougher). 16 is the only natural number that can be represented as x^y and y^x, where x, y are distinct. Who woulda thunk?

An irrational number raised to an irrational power can be rational (unlike probably a lot that I have mentioned in this post). Try proving that.

Every natural number in the world has a multiple that comprises all the digits appearing at least once each.

Did you know that two triangles can have 3 angles equal and 2 sides equal and still be not-congruent to each other? A quadrilateral can have a pair of sides equal and a pair of sides parallel and still not be a parallelogram. I think it is easier to be this quadrilateral than to be those two triangles.

Do you know that we put our kids through 15 years of school education without them discovering most of these facts? Let me stop right there. I can sense a rant against the Education system coming through the system.

"An irrational number raised to an irrational power can be rational (unlike probably a lot that I have mentioned in this post). Try proving that. " - B, I still remember the proof that you spelt out while riding on the TVS Champ... or was it TVS 50? sqrt(2)^sqrt(2)^sqrt(2)

ReplyDeleteSome TVS variant da. Dont remember which one. I remember this question from the recent conversation we had. That was the trigger for me to try to capture this. Fundooo idea, really.

ReplyDelete