Rational Numbers

Rational NumbersRational numbers can be expressed in the form where a and b are integers. b

a

Rational NumbersRational numbers can be expressed in the form where a and b are integers. b

a

Irrational Numbers

Rational NumbersRational numbers can be expressed in the form where a and b are integers. b

a

Irrational NumbersIrrational numbers are numbers which are not rational.

All irrational numbers can be expressed as a unique infinite decimal.

irrational is 2 Prove ..ge

irrational is 2 Prove ..ge“Proof by contradiction”

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a4by divisible is2by divisibleisthat squareAny

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a4by divisible is2by divisibleisthat squareAny

4by divisible bemust Thus 2a

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a4by divisible is2by divisibleisthat squareAny

4by divisible bemust Thus 2a2 2 4b k where k is an integer

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a4by divisible is2by divisibleisthat squareAny

4by divisible bemust Thus 2a2 2 4b k where k is an integer2 2b k

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a4by divisible is2by divisibleisthat squareAny

4by divisible bemust Thus 2a2 2 4b k

factorcommon a havemust and 2by divisibleboth are and So 22 ba

where k is an integer2 2b k

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a4by divisible is2by divisibleisthat squareAny

4by divisible bemust Thus 2a2 2 4b k

factorcommon a havemust and 2by divisibleboth are and So 22 baHowever, a and b have no common factors

where k is an integer2 2b k

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a4by divisible is2by divisibleisthat squareAny

4by divisible bemust Thus 2a2 2 4b k

factorcommon a havemust and 2by divisibleboth are and So 22 baHowever, a and b have no common factors

rationalnot is 2so

where k is an integer2 2b k

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a4by divisible is2by divisibleisthat squareAny

4by divisible bemust Thus 2a2 2 4b k

factorcommon a havemust and 2by divisibleboth are and So 22 baHowever, a and b have no common factors

rationalnot is 2soirrational is2

where k is an integer2 2b k

irrational is 2 Prove ..ge“Proof by contradiction”

rationalis 2 Assume

ba

2 where a and b are integers with no common factors

ab 2222 ab

2by divisible bemust Thus 2a4by divisible is2by divisibleisthat squareAny

4by divisible bemust Thus 2a2 2 4b k

factorcommon a havemust and 2by divisibleboth are and So 22 baHowever, a and b have no common factors

rationalnot is 2soirrational is2 Exercise 2B; 2a, 3, 4 (root 3), 8, 14*

where k is an integer2 2b k