so
on.
irrational
irrational
(i.e.,
(i.e.,surds).
surds).InInother
otherwords,
words,square
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rootsofofpositive
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surds.
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Thesame
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numberswill
willalso
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and
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Pure so
Mathematics
[B]
Do
2
2
you think that it is possible to raise an irrational number to an irrational power and get a
2 2
2 rational
[B]
[B] 2answer?
2
you
you
think
think
that
that
itis is
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raise
anan
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irrational
number
number
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irrational
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and
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itit is
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and
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rational
rational
answer?
answer?
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answer?
2
2
Suppose for example that a = 2 and b = 2 . Perhaps 2 is irrational (in fact it is, but
It Itisclever
ispossible.
possible.
the
thing is that you don’t need to know whether it is or isn’t).
2 2
2 2
Suppose
andb b= = 2 .2Perhaps
. Perhaps 2 2 is isirrational
irrational(in(infact
factit itis,is,but
but
Supposeforforexample
examplethat
thata a= = 2 2 and
thetheclever
cleverthing
thingis isthat
thatyou
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don’tneed
needtotoknow
knowwhether
whetherit itis isororisn’t).
isn’t).
2
Is 2 irrational?
2 2
irrational?
IsIs 2 2 irrational?
No
Yes
NoNo
Yes
Yes
Then consider  2

Then we have an example of
irrational irrational = rational , which
answers
theanproblem.
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Then
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have
anexample
exampleofof
irrational
irrational
= rational
= rational
, which
, which
irrational
irrational
answers
answersthetheproblem.
problem.
2
2
2
 . This is


2 2
equalThen
toThen
= consider
2consider
= 2 , which
 2 22 is
2 obviously
. Thisis isnot
  . This
 
irrational. So we have
  an
 example of
irrational
2 2
,iswhich
answers
irrational
,2which
, which
isobviously
obviously
not
not
equal
equaltoto= = 2 2==2=rational
the
problem.
irrational.
irrational.SoSowewehave
haveananexample
exampleofof
irrational
irrational
= rational
= rational
, which
, whichanswers
answers
irrational
irrational
< RH box, line 2: Pse deletethe
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theproblem.
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So either way the statement is proved.< <RH
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This
clever
linethe
of
reasoning
an
example of a ‘non-constructive proof’. Often proving that
SoSoeither
eitherway
way
thestatement
statementisisisproved.
proved.
e
numbers are irrational is very difficult. No-one knows whether π π , ee or , for instance, are
π Often
This
Thisclever
cleverline
lineofofreasoning
reasoningis isananexample
exampleofofa ‘non-constructive
a ‘non-constructiveproof’.
proof’.Often
proving
provingthat
that
irrational.
ee
e
or , for
, forinstance,
instance,areare
numbers
numbersareareirrational
irrationalis isvery
very
difficult.
difficult.No-one
No-oneknows
knowswhether
whetherπ ππ,π e, e eor
ππ
irrational.
irrational.
Folding
a cube root
•Fold a square piece of paper into three equal strips, parallel to the edge.
(There are proper origami ways of doing this – find out about Haga’s Theorem.)
•Then make the fold shown
(the dashed
lines show
1

3
the folds into thirds.)
a


[7]
[7][7]
a =32
•Now show that .
b
h
b
x
6
Ideas for Sixth-Form Mathematics