# 11x1 t02 02 rational & irrational (2011)

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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 ..geProof by contradiction

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factors

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

4by divisible is2by divisibleisthat squareAny

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

2 2 4b k where k is an integer

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

2 2 4b k where k is an integer2 2b k

irrational is 2 Prove ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

2 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 ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

2 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 ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

2 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 ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

2 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 ..geProof by contradiction

rationalis 2 Assume

ba 2 where a and b are integers with no common factorsab 2

222 ab 2by divisible bemust Thus 2a

4by divisible is2by divisibleisthat squareAny 4by divisible bemust Thus 2a

2 2 4b k

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

where k is an integer2 2b k

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