Download Irrational numbers

Few more elegant proofs
Irrational numbers1
R. Inkulu
http://www.iitg.ac.in/rinkulu/
1
objective of this lecture is not to understand the properties of rational and irrational
numbers; instead the focus was on learning how to prove few propositions together with
learning few summation techniques
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Definition
A real number that can be expressed as ba with b 6= 0 is termd as a rational
number. A real number that is not rational is said to be an irrational.
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Few properties
• Every rational number can be expressed as a finite continued fraction
(a0 +
1
a1 +
1
...+ a1
n
), whose integer coefficients ai can be determined by
applying the Euclidean algorithm
• The decimal expansion of an irrational number never repeats nor
terminates, unlike a rational number
• Almost all real numbers are irrational: rational numbers are countable
whereas the real numbers are uncountable
— proved later
• Rational numbers form a field
— not proved in class
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√
2 is irrational
We have seen various proofs of this stmt with • proof by contradiction
• proof by infinite descent
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√
k is irrational, if it is not integral
We have seen various proofs of this stmt with • proof by contradiction
• proof by infinite descent
• proof using fundamental theorem of arithmatic
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Golden ratio is irrational
golden ratio φ is a constant:
√
1+ 5
2
if φ is rational then 2φ − 1 =
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√
5 is rational — reaching contradiction
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lg2 3 is irrational
proof by contradiction
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√
√
2
2
is irrational
non-constructive existence proof
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e is irrational
if e =
a
b
for integers a and b > 0, then n!be = n!a for every n ≥ 0; but
bn!e
= bn!((1 +
1
1!
+
1
2!
+ ... +
1
n! )
1
+
+ ( (n+1)!
1
(n+2)!
+ . . .))
= (integral) +n!(non-integral) for sufficiently large n
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e2 is irrational
if e2 =
a
b
for integers a and b > 0, then n!be = n!ae−1 for every n ≥ 0; but
n!ae is just a bit smaller than an integer and n!be is a bit larger than an
integer
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e4 is irrational
— additional reading
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