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e is not a quadratic irrational
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(Proof)
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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.
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"e is not a quadratic irrational" is owned by mathcam. [ full author list (3) | owner history (1) ]
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(view preamble)
Cross-references: completes, absolute value, terms, fix, contradiction, converges, Taylor series, coefficients, integer, polynomial, root, quadratic extension, irrational
There is 1 reference to this entry.
This is version 8 of e is not a quadratic irrational, born on 2003-11-21, modified 2005-03-18.
Object id is 5426, canonical name is EIsIrrational.
Accessed 4085 times total.
Classification:
| AMS MSC: | 26E99 (Real functions :: Miscellaneous topics :: Miscellaneous) | | | 11J72 (Number theory :: Diophantine approximation, transcendental number theory :: Irrationality; linear independence over a field) |
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Pending Errata and Addenda
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