e is not a quadratic irrational
We wish to show that is not a quadratic irrational, i.e. is not a quadratic extension of . To do this, we show that it can not be the root of any quadratic polynomial with integer coefficients.
We begin by looking at the Taylor series for :
This converges for every , so and . Arguing
by contradiction, assume for integers
, and . That is the same as .
Fix , then and , . Consider
Since for , the first two terms are integers. So the third term should be an integer. However,
is less than by our assumption that . Since there is only one integer which is less than in absolute value
, this means that for every sufficiently large which is not the case because
is not identically zero. The contradiction completes the proof.
Title | e is not a quadratic irrational |
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Canonical name | EIsNotAQuadraticIrrational |
Date of creation | 2013-03-22 14:04:06 |
Last modified on | 2013-03-22 14:04:06 |
Owner | mathcam (2727) |
Last modified by | mathcam (2727) |
Numerical id | 11 |
Author | mathcam (2727) |
Entry type | Proof |
Classification | msc 11J72 |
Classification | msc 26E99 |
Related topic | EIsIrrationalProof |
Related topic | ErIsIrrationalForRinmathbbQsetminus0 |
Related topic | EIsTranscendental |