The Real Numbers (Theorems)
Original Notes adopted from November 6, 2001 (Week9)
© P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong
Prove
√
2 +
√
3 is irrational
Suppose
√
2 +
√
3 rational
√
2 +
√
3 = m/n
√
2 = m/n 
√
3
2 = m
2
/n
2
– 2 m/n
√
3 + 3
2m/n
√
3 = m
2
/n
2
+ 1
√
3 = n/2m (m
2
/n
2
+ 1)
But
√
3 irrational.
∴
Contradiction.
Extra problem : Is
√
2 +
√
3 +
√
5 rational?
Several ways of constructing real numbers from the rationals. We'll use
"Dedekind cuts".
Q = set of rational numbers
Each real number will be a set of rational numbers.
It's like the set of rationals less than the number.
0
√
2
Eg.
√
2 = { p
∈
Q: p < 0 or p
2
< 2 }
Definition: A real number x is a subset of Q such that:
1) x
≠
∅
, x
≠
Q (set of all Q)
2) if p
∈
x & s
∈
Q with s < p, then s
∈
x.
3) There is no largest number x.
Eg. 3 = { p
∈
Q: p < 3 }
To each rational number m/n, associate the set {p
∈
Q: p < m/n } = m/n x
≤
y if x
⊂
y x < y if x
⊂
y & x
≠
y.
√
2 = { p
∈
Q: p < 0 or p
2
< 2 }
Is
√
2 a real number?
√
2
≠
∅
(eg. 1
∈
√
2 )
√
2
≠
Q (eg. 7
∉
√
2 )
Show p
∈
√
2 & q < p
⇒
q
∈
√
2 Two Cases:
1)
if p
≤
0
if q < p, then q < 0, so q
∈
√
2
2)
if p > 0 & p
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 Spring '10
 Applebaugh
 Math, Real Numbers, Rational number, Irrational number

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