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

Note by Pi Han Goh
1 year, 2 months ago

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

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

I calculate it using a website.

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

- 7 months, 3 weeks 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...

- 1 year, 1 month ago

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

- 1 year, 1 month 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.

- 1 year, 1 month ago

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.

- 1 year, 1 month ago

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!

- 1 year, 2 months ago