# Tetration

I stumbled over this problem, thinking it to be just another tricky math problem:

$\displaystyle \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}$ = ?

Trying out by regular method of solving this problem, i.e removing one iteration in this infinite exponentiation, we get

$\displaystyle (\sqrt{2})^x = x,$

Now the above equation has two solutions which we see intuitively. x=2 as well as x=4 seem to satisfy the equation. This too doesn’t take me by surprise because in such situations one of them will be right and the other wrong because the LHS in the problem is a well defined real number and has to be unique. But, I recollected,

Any number greater than 1, raised to a power greater than 1 would always grow exponentially. Hence, an infinite exponentiation(tetration) of such number would yield infinity as the result.

I felt it to be quite intuitive! But here, I ended up with 2 solutions and both appear to be converging, while I expected it to be diverging. After a bit of googling I hit the right terms of Tetration and Infinite Power Tower Problem. I found out to my dismay, that it so happens,

Infinite tower power $x^{x^{\cdot^{\cdot^{\cdot}}}}$ is continuous as a function, whose domain is the interval $[e^{-e}, \sqrt[e]{e}]$ and whose range is $[e^{-1},e]$.

I’m taken aback by the surprises math has in store to offer me. I thought I knew basic math and believed in some things as a thumb-rule which were so intuitive that they could not be denied. Its these mysteries, that drive us towards understanding things. Lesson learnt is,

However intuitive a thing might be, if you cannot prove it, you better not believe in it!

Jeremy Kun has a nice post, False proof 2=4 on validity of the solution and a link on detailed proof of the same.

# The Sandwich Number

I came across this interesting fact while browsing one of the mathematical books. It is the sandwich number as termed by Fermat. A sandwich number is one, which is the only number which lies in between a perfect-square and a perfect-cube. One example of it is the number 26. It lies between 25(which is 5 squared) and 27(which is 3 cubed). The astonishing thing proposed by Fermat was 26 is the ONLY sandwich number!! And he even put forward a proof in his usual style(scribbling in margins).

Although this fact appears simple, the proof might either be much complicated or a tricky one, as is the case with many number theory problems. If anyone comes across a proof to this problem, please enlighten me!

# The Second minutes

Mathematical concepts usually appear obscure and most try to shun away from them. But, once we trace back the history of time to see the growth of such abstract concepts we cant help appreciating them. Once such thing is the most-feared term ‘Calculus’ which has crept into our daily lives but we fail to recognize it. Here is a simple example: We daily use the term second (unit of time) but do not know why we call it so.

Here is an excerpt from the book “Calculus made easy” by Thompson which explains it:

There are 60 minutes in an hour,24 hours in a day,7 days in a week. There are therefore 1440 minutes in a day and 10080 minutes in a week. Obviously 1 minute is a small quantity of time compared with a whole week. Indeed, our forefathers found it small as compared with an hour and called it ‘one minute’ meaning a minute fraction-namely one-sixtieth of an hour. When they came to require still smaller subdivisions of time, they divided minute into 60 still smaller parts, which they called “second minutes” (i.e., small quantities of second order of minuteness). Gradually, this second order of smallness became seconds. But very few know why they are called so.