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e is irrational
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| Pett |
If e were rational, then e = n/m for some integers m, n. So then 1/e = m/n. But the series expansion for 1/e is
1/e = 1 - 1/1! + 1/2! - 1/3! + ...
Call the first n terms of this alternating series S(n). How good is this approximation to e? Well, the error is bounded by the next term of the alternating series:
0 < | 1/e - S(n) | = | m/n - S(n)| < 1/(n+1)!
But multiplying through by n!, you will see that
0 < | integer - integer | < 1/(n+1) < 1.
But there is no integer strictly between 0 and 1, so this is a contradiction; e must be irrational. |
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| RJT |
| **Insert rational e-pill joke here** |
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| idoru |
| I finished the first sentence and wanted to shoot myself. I'm not even joking, my stress level just shot through the roof. God, I hate math. |
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| RJT |
| quote: | Originally posted by idoru
I finished the first sentence and wanted to shoot myself. |
You too? |
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| Sushipunk |
| quote: | Originally posted by idoru
I finished the first sentence and wanted to shoot you in the face. I'm not even joking, my stress level just shot through the roof. God, I hate your post. |
First time I've ever done this, but...
FIXED. |
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| Nsonic |
ecstacy is a number? :conf:
EDIT>
e
| quote: | | It is known that e is irrational (proof) |
i see.. |
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| Subey |
| quote: | Originally posted by Pett
If e were rational, then e = n/m for some integers m, n. So then 1/e = m/n. But the series expansion for 1/e is
1/e = 1 - 1/1! + 1/2! - 1/3! + ...
Call the first n terms of this alternating series S(n). How good is this approximation to e? Well, the error is bounded by the next term of the alternating series:
0 < | 1/e - S(n) | = | m/n - S(n)| < 1/(n+1)!
But multiplying through by n!, you will see that
0 < | integer - integer | < 1/(n+1) < 1.
But there is no integer strictly between 0 and 1, so this is a contradiction; e must be irrational. |
Of course i'm going to use the colour red as an example. It's at the end of the spectrum so the rational of clipping it doesn't require a significant leap to link the two data points into a single cohesive logical unit.
Toronto! Ya give em the CN tower and they mistake it for the Peace Tower. |
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| Bidor |
| that. E is my friend. :gsmile: |
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| Krypton |
E = MC^2
Two Mc's equals E. E MC's. Yo. |
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| Krypton |
| quote: | Originally posted by Bidor
1+1=4? |
not even imaginary. just wrong. |
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