|
log 2
is irrational |
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What
is a rational number ? An rational
number is a number, n, which can be written in the form where p and q are integers, q ¹ 0 and the H.C.F. of p and
q is equal to 1. In
short, a rational number can be written in a fraction of two integers and an irrational
number cannot. |
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Short
proof of “log 2 is irrational” Assume
that log 2 is rational, that is, where
p, q are integers. Since
log 1 = 0 and log 10 = 1, 0 <
log 2 < 1 and therefore p
< q. From
(1), Now,
it can be seen that the L.H.S. is even and the R.H.S. is odd. Hence
there is contradiction and log
2 is irrational. |
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For
more serious learners Square root of
2 is also irrational. There are many proofs. You may study a few of them
by clicking here. |
