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

Note by Pi Han Goh
3 months ago

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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 · 2 months, 4 weeks ago

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@Steven Perkins Even 22/7 is close to pi but not to that much accuracy and precision as this number is close to 'e' Terry Chadwick · 2 months, 4 weeks ago

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@Terry Chadwick 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 · 2 months, 3 weeks ago

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@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. Terry Chadwick · 2 months, 3 weeks ago

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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! Terry Chadwick · 3 months ago

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