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In Reply to: Sorry? posted by Victor Khomenko on July 25, 2000 at 19:02:15:
Victor,Here's one for you:
Without use of a calculator, decide which of the following two numbers is bigger: e raised to the pi (e^(pi)) or pi raised
to the e power ((pi)^e) ? It is permissible to ask a Harvard/Cornell
grad for help.The Concorde tragedy is another reminder to us that ultimately we are quite fallible.
...my intuitive feel is always to favor the higher power. I have to admit that I checked my intuition with an HP-41X.Besides, I was the one to mention that story about the chess inventor.
I presume you see the parallel.
Sorry, don't know the story about the inventor of chess....would enjoy hearing it.on the 'powers problem:' please see Gnat's proof below. A more informal version of Gnat's argument would involve comparing the graphs of x^e and e^x (for positive x) --- these graphs intersect only once in the first quadrant, at x = e.
What is that movie where a lady mathematician has a brief affair with Michael Douglas?
Hi, Victor> > >
...my intuitive feel is always to favor the higher power.
< < <Though I agree that e^pi > pi^e, the higher power per se
doesn't guarrantee nothing, 2^3 is LESS than 3^2.
My proof is as follows:e^pi ??? pi^e
ln( e^pi ) ??? ln( pi^e )
pi ??? e * ln( pi )
pi - e * ln( pi ) ??? 0
Now let's consider it as the function
f( x ) = x - e * ln( x ) when x = piderivative is
f'( x ) = 1 - e / x,f( x ) minimum is when f'( x ) = 0, when x = e,
min value is
f( e ) = e - e * ln( e ) = 0Well, that means
x - e * ln( x ) > 0, when x not= epi - e * ln( pi ) > 0
pi > e * ln( pi )
ln( e^pi ) > ln( pi^e )
e^pi > pi^e
Hope my deduction is correct. :)
regards, gnat
Great! This is your second award in just few days.And of course this is the right way of approaching it if you need a precise answer - and I would not be able to do it this fast (although I believe you would certainly enjoy some time with Leon and listen to his Kolyma stories. His family history is one from Shalamov...).
My presumption was, however, that the original question was more in line of the typical American multiple choices question, where you don't have time to go through equations but need to evaluate few alternatives quickly. So you would look perhaps at three cases: =, <, and > . And yes, I did consider that 2^3 vs. 3^2 as well (that one naturally crosses one's mind, I think). I however decided that the deiviation from the center point where a=b so a^b=b^a was still small and the function monotonous. That was enough for me to feel very secure about that decision in just few seconds. Not certain, but secure.
BTW, there is a whole big school that teaches quick decision making under stress. I took numerous classes while still at HP and while I found them mostly boring, they had some interesting things in them. It is about making decisions without having enough information available. Of course you can only evaluate such approaches statistically and not on case-by-case basis.
Leon is also a man of crystal-clear thinking and bear-trap intuition. I had always enjoyed watching him to tackle a completely new task - something he had no slightest idea about. Knowing a lot of fundamental things he would always quickly arrive at the first approximation answer just standing by the coffee machine. In majority of those cases he was right on the money. I will certainly never be able to approach his speed and vision, but I know what my fifteen years of working with him had done for me. Mentors like that don't come along too often.
PS. Got in touch with folks at Souvenir - Russian music store, and again feel that I could do without it. Hope to get some CD's soon, will let you know.
Hi, Victor
This darn problem already took a few hours of my life.
And I'm still thinking about it.As I understand, your fast considerations are true only
if that center point is equal or bigger than e. Otherwise
it seems to be wrong, sorry. :) I didn't calculate, but
I SWEAR that say, 2.69^2.71 is LESS than 2.71^2.69
If honestly, I'd not be able to make a quick decision for
this question. Well, I tried to evaluate it "by fingers"
first, but after a few attempts I gave up and decided just
to do all this math. I'd like to take these decision
classes, since it's my weak point INDEED. Usually I think,
rethink, check, recheck and, after a few hours of doubts,
release the answer. ;-)
Are Vopli Vidoplyasova available on your Souvenir? Their
latest album also seems to be OK. Just spell: "Oj buli,
buli na seli".
regards, gnat
You must have one BIG dacha...***This darn problem already took a few hours of my life.
And I'm still thinking about it.Same here.
***As I understand, your fast considerations are true only
if that center point is equal or bigger than e.I didn't know that. I knew quickly that getting close to 1 was dangerous - 1^64 vs. 64^1 - hmmm... But I didn't see the e as the break point, of course. However both numbers being close to some center point gave me some assurance. Enough to bet my dollar.
***Otherwise
it seems to be wrong, sorry. :) I didn't calculate, but
I SWEAR that say, 2.69^2.71 is LESS than 2.71^2.69No questions, such quick evaluation is not always going to work. But as I said - often the decisions in life call for one being "secure" rather than right. Especially when the information is not present.
Since this subject interests me, I would be very curious if someone could provide another way of doing quick estimation on a fly.
***If honestly, I'd not be able to make a quick decision for
this question. Well, I tried to evaluate it "by fingers"
first, but after a few attempts I gave up and decided just
to do all this math.I suspect this is all in the mindset. You are obviously trained in exact science and are usually able to arrive at precise answer (and very effectively, thank you). That is great, but has a small byproduct that smart people had discovered some time ago - absence of such firm ground sometimes stops the process. But in many cases the process can not tolerate being stopped - something MUST be done and such cases call for a different approach. We all can come up with many situations like that.
I have to admit that the first time I got exposed to this concept of decision making as science I was very fascinated. Many professions - astronauts, nuclear plant operators, commandos, even government officials are trained in this. Most people never get to see it.
This resonated in me for some reason.
***I'd like to take these decision
classes, since it's my weak point INDEED. Usually I think,
rethink, check, recheck and, after a few hours of doubts,
release the answer. ;-)Your mind is very analytical. I mentioned before that you remind me of Leon - the endless resource of wisdom. He is a tremendous scientist. He graduated from the Bonch-Bruevitch, but never actually worked in Russia as an engineer - always the guy with pages of formulas, but no circuits. When he came here he had to start working as a EE and quickly had become the best EE brain in our HP division, a guy with line of people waiting to get his advice on anything, from noise analysis to trading stock to something he had never even heard about before - gas chromatography, of all fields. He then spent time doing fancy analytical algorithms, creating several interesting things along the way. And then - he jumped into the phisycal chemstry both feet. In few short years he had become a world leading scientist in several areas of gas chromatography. When HP finally decided that they were paying him too much and that some young idiot could do all the same things at 1/3 of his pay, he retired and is now one of the most respected specialists in the world. He goes to the conferences and does consulting.
Well, maybe I am just using this opportunity as an excuse to talk about him a bit.
Anyhow, what had always surprised me in him was his uncanny ability to see the right thing through any smoke and ambiguity. In subjects that he knew nothign about. No, he was not right in every case, but if someone like him was in charge of this coutry you could be darn sure his net balance would be beyond positive. He was full of simple rule-of-thumbs that he had developed over the years and some of them sounded outright stupid when you first heard them - but you could take them all straight to the bank - they were bullet-and-rocket-proof. I am still using as many of them as I was able to remember.
Which reminds me... I need to send him $300...
Anyway, a VERY fascinating field....
***Are Vopli Vidoplyasova available on your Souvenir? Their
latest album also seems to be OK. Just spell: "Oj buli,
buli na seli".With the archive being down, I forgot about this one. I had ordered a bunch of others, but boy, is interacting with those people funny! Like buying pork on the Odessa market - "Are you sure you want this one? Try this one instead! But I want this one! Leva, pidjak wants this one! Tell him he DOESN'T want it! He wants that one with the red cover!" and on, and on... you get the picture... She didn't have Alsu that I ordered, but had some disco remixes, and - but of course! - that was much better and I absolutely must have it!
So I will check on Vopli.
Getting too long, so see you later
Hi, Victor
'Tsa-gi' is Central AeroHydrodynamic Institute, dad is just
a normal :) soviet engineer.
> > >
You must have one BIG dacha...
< < <Yep, but it was NOT as big for my great-granfather family.
> > >
***As I understand, your fast considerations are true only
if that center point is equal or bigger than e.I didn't know that. I knew quickly that getting close to
1 was dangerous - 1^64 vs. 64^1 - hmmm... But I didn't see
the e as the break point, of course. However both numbers
being close to some center point gave me some assurance.
Enough to bet my dollar.
< < <It's all in derivatives to me.
f(x)=a^x, g(x)=x^a,
f'(x)=ln(a)*a^x, g'(x)=a*x^(a-1)
When x = a, we get f'(a)=ln(a)*a^a, g'(a)=a*a^(a-1)=a^a
f'(a) is less than g'(a) when ln(a) is less than 1, i.e
when a is less than eThis determines functions behaviour near the intersection
point, including which one increases faster.
> > >
***Otherwise
it seems to be wrong, sorry. :) I didn't calculate, but
I SWEAR that say, 2.69^2.71 is LESS than 2.71^2.69No questions, such quick evaluation is not always going to
work. But as I said - often the decisions in life call for
one being "secure" rather than right. Especially when the
information is not present.
< < <Yes. Another reason for it is that even these sloooooooow
decisions aren't error-free. At least IME. :) And when time
is restricted, I prefer ANY decision over the brain-lock.
> > >
Since this subject interests me, I would be very curious if
someone could provide another way of doing quick estimation
on a fly.
< < <That's not to me. I'm also familiar with a few guys like your
friend Leon. And I always admired their ability to provide not
just proper, but ELEGANT solutions - thing impossible to me.
> > >
***If honestly, I'd not be able to make a quick decision for
this question. Well, I tried to evaluate it "by fingers"
first, but after a few attempts I gave up and decided just
to do all this math.I suspect this is all in the mindset. You are obviously trained
in exact science and are usually able to arrive at precise answer
(and very effectively, thank you). That is great, but has a small
byproduct that smart people had discovered some time ago - absence
of such firm ground sometimes stops the process. But in many cases
the process can not tolerate being stopped - something MUST be done
and such cases call for a different approach.
We all can come up with many situations like that.I have to admit that the first time I got exposed to this concept
of decision making as science I was very fascinated.
Many professions - astronauts, nuclear plant operators, commandos,
even government officials are trained in this.
Most people never get to see it.This resonated in me for some reason.
< < <Gee I NEED such a training!
> > >
Which reminds me... I need to send him $300...
< < <Hey Victor, can you pass to Leon this related problem:
when a > 1 and a not equal e, a^x and x^a have ANOTHER
intersection besides x=a. How to find it analytically???? :)
I've spent last 3 or 4 days masturbating my brain over it,
but I still "can't get no satisfaction", darn.
regards, gnat
a^x=x^a
x*ln(a)=a*ln(x)
x/a=ln(x-a)
(e^x)/(e^a)=e^(ln(x-a))
(e^x)/(e^a)=x-a
(e^x)*(e^(-a))=x-a
e^(-a*x)=x-a
now take da/dx for both sides
(-a)*e^(-a*x)=1
e^(-a*x)=-a^-1
ln(e^(-a*x))=ln(-a^-1)
-a*x=-ln(-a)
a*x=ln(-a)
x=ln(-a)/a
since a> =0, ln(-a)=infinity
therefore x=infinity.
checking our work
a^x=x^a
a^infinity=infinity^a for all a> 1
yes, x=infinity is other solution.
make that take d/dx NOT da/dx
'Tsa-gi' is Central AeroHydrodynamic Institute, dad is just
a normal :) soviet engineer.I presume - retired now? Is TsAGI still strong? I believe they have lost a lot of their personnel.
< < <
It's all in derivatives to me.
f(x)=a^x, g(x)=x^a,
f'(x)=ln(a)*a^x, g'(x)=a*x^(a-1)
When x = a, we get f'(a)=ln(a)*a^a, g'(a)=a*a^(a-1)=a^a
f'(a) is less than g'(a) when ln(a) is less than 1, i.e
when a is less than e***This determines functions behaviour near the intersection
point, including which one increases faster.
Yes, absolutely true. The simple graphical analysis doesn't tell you where they intersect, just that they do.> > >
***Otherwise
it seems to be wrong, sorry. :) I didn't calculate, but
I SWEAR that say, 2.69^2.71 is LESS than 2.71^2.69No need, your formula tells it all.
***That's not to me. I'm also familiar with a few guys like your
friend Leon. And I always admired their ability to provide not
just proper, but ELEGANT solutions - thing impossible to me.
***I have to admit that the first time I got exposed to this concept
of decision making as science I was very fascinated.
Many professions - astronauts, nuclear plant operators, commandos,
even government officials are trained in this.
Most people never get to see it.***This resonated in me for some reason.
< < <***Gee I NEED such a training!
All you would need (and I am not for a second suggesting this - BTW!) would be few weeks with my wife. She would put relentless pressure on you - C'mon now, decide already! She firmly believes that all major decisions (and the resulting responsibility for the wrong ones) must fall on my shoulders. She will go with most of them but reserves the right to crucify me for bad ones. Stalin - Zhukov relationship, if you will. Freedom to make decisions for as long as you make good ones...
I'have managed to stay from the family Gulag for this long...
***Hey Victor, can you pass to Leon this related problem:***when a > 1 and a not equal e, a^x and x^a have ANOTHER
intersection besides x=a. How to find it analytically???? :)
I've spent last 3 or 4 days masturbating my brain over it,
but I still "can't get no satisfaction", darn.I gave it to him. I don't see much of him anymore, unfortunately, but just as random chance would dictate, he stopped by my office yesterday, right after you and I talked here.
BTW - explaining things like that to him is NOT trivial - he always starts asking questions that seem orthogonal to your problem.
I suspect we may get kicked out of the Film forum soon.
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