How would one show that e^pi is greater than pi^e without using a calculator?
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Hxc said:Unless I'm missing something (only doing GCSE maths currently), e stands for any number, correct?
say e = 2
2^pi = 8.82497783
pi^2 = 9.8696044
Which means that either your statement is incorrect, or that I do not know what e is
Soldato
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e is the base for a natural logarithm (log base e or ln)
According to Wikipedia, e to 20 decimal places (it's irrational) is 2.71828182845904523536.
I just tried to prove this, but actually, I showed that e^pi is less than pi^e, so I seemed to have failed.
According to Wikipedia, e to 20 decimal places (it's irrational) is 2.71828182845904523536.
I just tried to prove this, but actually, I showed that e^pi is less than pi^e, so I seemed to have failed.
DaveF said:
Hmmm,
I have a feeling there may be something more elegant.
IF this was a interview question, then they can't expect ppl to do that on the spot, needs a pen and paper.
My first instinct is to take logs of both sides.
but then showing pi > e ln pi isn't straight forward as I thought
sid
sid said:
I had interviews for Engineering at both Cambridge and Oxford and was asked similar problem solving questions, you're allowed a pen and paper.
The interviews aren't "sit on a chair in the middle of a room and be interrogated," you usually sit down at a table with one or two people and go through some questions.Calder said:I had interviews for Engineering at both Cambridge and Oxford and was asked similar problem solving questions, you're allowed a pen and paper.
Lol I've been to one of these interview as well
They make you think outside the box, something clearly I wasn't very good at
Back to the problem, I think thats a toughie without perhaps a hint.
sid
not done by me but:
Consider the function f(x) = e^x - x^e. Its derivative is
f'(x) = e^x - e*x^(e-1)
f'(x) = 0 will have at most two solutions since f'(x) = 0 implies
e^x = e*x^(e-1)
and it is easily seen that an exponential function and a power
function have at most two points of intersection. Observe that
f'(1) = 0 and f'(e) = 0, so x = 1 and x = e are the two points of
intersection.
In general, an exponential function will grow to infinity faster than
a power function, so f'(x) >= 0 for x > e. This means that f is an
increasing function when x > e. Also notice that f(e) = 0. Thus,
since Pi > e and f is increasing when x > e, we have f(Pi) > 0.
Thus f(Pi) = e^Pi - Pi^e > 0. This implies that e^Pi is greater than
Pi^e.
Consider the function f(x) = e^x - x^e. Its derivative is
f'(x) = e^x - e*x^(e-1)
f'(x) = 0 will have at most two solutions since f'(x) = 0 implies
e^x = e*x^(e-1)
and it is easily seen that an exponential function and a power
function have at most two points of intersection. Observe that
f'(1) = 0 and f'(e) = 0, so x = 1 and x = e are the two points of
intersection.
In general, an exponential function will grow to infinity faster than
a power function, so f'(x) >= 0 for x > e. This means that f is an
increasing function when x > e. Also notice that f(e) = 0. Thus,
since Pi > e and f is increasing when x > e, we have f(Pi) > 0.
Thus f(Pi) = e^Pi - Pi^e > 0. This implies that e^Pi is greater than
Pi^e.
Not that tough (though of course they'll give you hints as necessary).sid said:
As you say, an obvous step is to take logs, so you're comparing
pi with e ln pi. So you want to compare e with pi / ln pi.
At which point looking at f(x) = ln(x)/x isn't that big a leap.
Then f(x) has a maximum at x = e, where f(x) = 1/e. So 1/e < ln pi / pi, so e > pi / ln pi.
The answer I gave is with "benefit of hindsight" to make it shorter, but the above shows how you might find it yourself.
DaveF said:
Your second line is very clever, I wouldn't have thought of that lol.
I can think maths, I guess thats how I ended up doing physics in some ways, its easier to think about.
BTW Its obvious because Exponentials pwn Polynomials lol
sid
Well when you raise to a power your answer grows exponentially. Since pi is greater than e, but not by a huge amount, you'd expect e^pi to bigger.PinkPig said:
But I've just realised this reasoning is even more crap than I thought it was since e and pi are too small for it to be as significant as I originally thought. Yeah so just ignore me, I don't know what I'm on about.