Which number is bigger? $e^7 - e$ or $ \frac{3758537274}{3435859} $?

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

I took the value of e till 6 decimal places and I observed that e was smaller than the rational no.

Though, it's a very close case and if we increase the value of decimal places of e, we might get closer look.

Unfortunately, my calculator isn't that advanced.

It's a great question and observation. I am eager to know the answer!

Kaushik Chandra - 4 years ago

They are the same to 17 digits: 1093.914876600000

So your calculator has to have more precision than that. I know which is larger, but don't have a helpful answer as to why, or why they are so close. I'm looking forward to further discussion...

Steven Perkins - 4 years ago

Even 22/7 is close to pi but not to that much accuracy and precision as this number is close to 'e'

Kaushik Chandra - 4 years ago

You can always construct rational numbers that are arbitrarily close to a given irrational number.

There are some specific cases where a rational number may have some basis in a series expansion or something else interesting. I haven't figured anything special out in this case, but I don't have much experience doing something like that.

I hope we'll get some more insight here eventually.

Steven Perkins - 4 years ago

@Steven Perkins – And the larger the number of digits in a rational number the more accurate it's till a particular decimal place.

I agree that we will learn something interesting here. I am thankful to Mr. Pi Han Goh for posting this wonderful relationship.

Kaushik Chandra - 4 years ago

$\large e^7-e= \boxed{1093.9148765}999995540283599508167687699444618877399$

$\large \frac{3758537274}{3435859}= \boxed{1093.9148766}000001746288191686562225050562319350125$

I calculate it using a website.

Here I can see the second one is a bit greater than first.

Md Mehedi Hasan - 3 years, 6 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Which number is bigger? \(e^7 - e\) or \( \frac{3758537274}{3435859} \)?

Comments