Integer part

idontknow · January 18th, 2019, 07:49 AM

How can I find the integer part of $\displaystyle e^e \;$? (without calculator)
e - Euler constant
What about approximation?

First, I tried one example for $\displaystyle 2e$ and $\displaystyle e^2$, to see whether it is possible.
$\displaystyle 3>e>\frac{5}{2}\;$ ,then multiply by 2
6>2e>5 so integer part of 2e is 5.

skipjack · January 18th, 2019, 11:27 PM

How much basic arithmetic is acceptable, given that use of a calculator isn't allowed? Can you assume that e = 2.718... or does that have to be proved as well?

idontknow · January 19th, 2019, 01:01 AM

By calculator ,integer part of $\displaystyle e^e$ is 15.
But integer part of $\displaystyle {2.7}^{2.7}$ is 14 , same for 2.71 .

skipjack · January 19th, 2019, 04:56 AM

$e^{2.71} >15$, but I think a lot of arithmetic is needed to prove that.

January 18th, 2019, 07:49 AM	#1
idontknow Senior Member Joined: Dec 2015 From: iPhone Posts: 480 Thanks: 73	Integer part How can I find the integer part of $\displaystyle e^e \;$? (without calculator) e - Euler constant What about approximation? First, I tried one example for $\displaystyle 2e$ and $\displaystyle e^2$, to see whether it is possible. $\displaystyle 3>e>\frac{5}{2}\;$ ,then multiply by 2 6>2e>5 so integer part of 2e is 5. Last edited by skipjack; January 18th, 2019 at 11:21 PM.
$idontknow is offline$

January 18th, 2019, 11:27 PM	#2
skipjack Global Moderator Joined: Dec 2006 Posts: 20,472 Thanks: 2039	How much basic arithmetic is acceptable, given that use of a calculator isn't allowed? Can you assume that e = 2.718... or does that have to be proved as well?
$skipjack is offline$

January 19th, 2019, 01:01 AM	#3
idontknow Senior Member Joined: Dec 2015 From: iPhone Posts: 480 Thanks: 73	By calculator ,integer part of $\displaystyle e^e$ is 15. But integer part of $\displaystyle {2.7}^{2.7}$ is 14 , same for 2.71 . Last edited by idontknow; January 19th, 2019 at 01:07 AM.
$idontknow is offline$

January 19th, 2019, 04:56 AM	#4
skipjack Global Moderator Joined: Dec 2006 Posts: 20,472 Thanks: 2039	$e^{2.71} >15$, but I think a lot of arithmetic is needed to prove that.
$skipjack is offline$

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