2. Rational Numbers
a
Rational numbers can be expressed in the form where a and
b are integers. b
3. Rational Numbers
a
Rational numbers can be expressed in the form where a and
b are integers. b
Irrational Numbers
4. Rational Numbers
a
Rational numbers can be expressed in the form where a and
b are integers. b
Irrational Numbers
Irrational numbers are numbers which are not rational.
All irrational numbers can be expressed as a unique infinite
decimal.
6. e.g . Prove 2 is irrational
“Proof by contradiction”
7. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
8. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
9. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
10. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
11. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
12. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
Any square that is divisible by 2 is divisible by 4
13. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
Any square that is divisible by 2 is divisible by 4
Thus a 2 must be divisible by 4
14. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
Any square that is divisible by 2 is divisible by 4
Thus a 2 must be divisible by 4
2b 2 4k where k is an integer
15. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
Any square that is divisible by 2 is divisible by 4
Thus a 2 must be divisible by 4
2b 2 4k where k is an integer
b 2 2k
16. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
Any square that is divisible by 2 is divisible by 4
Thus a 2 must be divisible by 4
2b 2 4k where k is an integer
b 2 2k
So a 2 and b 2 are both divisible by 2 and must have a common factor
17. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
Any square that is divisible by 2 is divisible by 4
Thus a 2 must be divisible by 4
2b 2 4k where k is an integer
b 2 2k
So a 2 and b 2 are both divisible by 2 and must have a common factor
However, a and b have no common factors
18. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
Any square that is divisible by 2 is divisible by 4
Thus a 2 must be divisible by 4
2b 2 4k where k is an integer
b 2 2k
So a 2 and b 2 are both divisible by 2 and must have a common factor
However, a and b have no common factors
so 2 is not rational
19. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
Any square that is divisible by 2 is divisible by 4
Thus a 2 must be divisible by 4
2b 2 4k where k is an integer
b 2 2k
So a 2 and b 2 are both divisible by 2 and must have a common factor
However, a and b have no common factors
so 2 is not rational
2 is irrational
20. e.g . Prove 2 is irrational
“Proof by contradiction”
Assume 2 is rational
a
2 where a and b are integers with no common factors
b
b 2a
2b 2 a 2
Thus a 2 must be divisible by 2
Any square that is divisible by 2 is divisible by 4
Thus a 2 must be divisible by 4
2b 2 4k where k is an integer
b 2 2k
So a 2 and b 2 are both divisible by 2 and must have a common factor
However, a and b have no common factors
so 2 is not rational
2 is irrational Exercise 2B; 2a, 3, 4 (root 3), 8, 14*