Rational numbers can be expressed as fractions with integer numerators and denominators. Irrational numbers cannot be expressed as fractions and instead have non-repeating decimal representations. The document proves that sqrt(2) is irrational by assuming it is rational, then showing this leads to a contradiction since the numerator and denominator would have a common factor.

1. Rational Numbers

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.

5. e.g . Prove 2 is irrational

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 2a

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 2a
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 2a
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 2a
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 2a
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 2a
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 2a
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 2a
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 2a
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 2a
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 2a
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 2a
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*