I learned my "One New Thing For The Day" from this interesting video, where a group from Numberphile unrolled a printout of Pi to the millionth decimal place on one mile of a runway... :Applause:

 

Sounds irrational *ducks*

Sounds irrational *ducks*

Speaking of "irrational..."
 
If you raise an irrational number to a rational power, it is possible to get something rational. For instance, raise Sqrt[2] to the power 2 and you'll get 2.
 
But what happens if you raise an irrational number to an irrational power? Can this ever be rational?
 
Hint: what is Sqrt[2]^{Sqrt[2]Sqrt[2]}

 

 

Hint: what is Sqrt[2]Sqrt[2]

1.6325....

I am sorry, all this talk about Pi has made me irrationally hungry. 

I like imaginary number myself B)

1.6325....

Um, not even close. The correct answer is...

 

2

 

The square root of any irrational number raised to the power of the square root of that same irrational number will always be a rational number!

Like all parents, the farmer wanted his son to have a better life than he had. And again, like most parents, he viewed sending the boy to college as the best way to accomplish this. So, the farmer scrimped and saved until he had enough money to send his son off to college to become an engineer.

 

The boy studied hard and was doing well but could hardly wait for the first school break to come home and see his parents.

 

The boy came home, walked in the house, and hugged his mother and father. After some small talk the father said "well, let's hear some of that there engineer schooling you been getting!"

 

The boy thought for a moment and proudly said "Well, I learned that the circumference of a circle is pi r squared"

 

The farmer took off his hat, scratched his head, and dropped his gaze to the floor.

He slowly looked up at his son and said:

 

"Damn it boy! Don't you know nuthin! I don't know what the hell a circumcised circle is, but any dang fool knows that pie are ROUND!!! I should've used that there college money to buy a new tractor!"

 2

Totally false.

if sqrt(2) raised to the power 2 equals 2 (we know that's true) then the same sqrt(2) raised to sqrt(2) can NEVER be 2.

I suggest you take a scientific calculator and actually compute it, there are online calculators that let you do it,

you can also double-check your result using logs alone. You can also start with some approximation, say sqrt(2) = 1.414 and

see what happens when you raise 1.414 to 1.414.

 

 

Hint: what is Sqrt[2]Sqrt[2]

 

I think you meant sqrt(2)^sqrt(2)^sqrt(2). sqrt(2)^sqrt(2) = 1.6325..., but sqrt(2)^sqrt(2)^sqrt(2) or 1.6325...^1.6325 = 2. You can do this on paper.

or 1.6325...^1.6325 = 2

1.6325^1.414 = 2, yes, I can accept it. Incomplete original notation, incomplete chain of powers.

I think you meant sqrt(2)^sqrt(2)^sqrt(2). sqrt(2)^sqrt(2) = 1.6325..., but sqrt(2)^sqrt(2)^sqrt(2) or 1.6325...^1.6325 = 2. You can do this on paper.

You are quite correct. That's what happens everytime I try to be clever; I get hoisted on my own petard because I was in a hurry and didn't check my message carefully enough. :blush:

• Theorem. There exist irrational numbers A and B so that A^B is rational.

Proof. We know that √2 is irrational. So, if A=√2 and B=√2 satisfy the conclusion of the theorem, then we are done. If they do not, then √2^√2 is irrational, so let A be this number. Then, letting B=√2, it is easy to verify that A^B=2 which is rational and hence would satisfy the conclusion of the theorem. QED.

 

Su, Francis E., et al. "Rational Irrational Power." Math Fun Facts. <http://www.math.hmc.edu/funfacts>

And now my brain hurts from having read all that. And it didn't help that it was a long day at work....

Pi r squared

Pie are square?

No, CORNBREAD are square! Pie are round!

 

 

Su, Francis E., et al. "Rational Irrational Power." Math Fun Facts.

 

Thanks Fr. Bill. You have forced me to recall what I have spent decades slowly not remembering bits at a time. Now I have got a head full of numbers and equations looking for some meaning or purpose again! I bet when I try to go to sleep tonight I'll be seeing Geometry, Trigonometry and Calculus problems to be solved instead of beautiful women and tropical beaches. Yep, . . . . thanks a whole bunch!

 

 

 

I get hoisted on my own petard

 

Was he not one of the Star Trek Captains? You have your own Captain?

 

Seriously, . . . . . . , good phrase usage. It has been a long time since I've heard / read that one.

 

Pi R sq, no, Pi R round - Cake R sq.

 

Regards,