Karnataka Secondary Education Examination BoardKarnataka SSLC

# Proofs of Irrationality

#### description

of Squareroot of 2

#### notes

Irrational numbers are numbers that cannot be written in the form of p/q, where p and q are integers and q ≠ 0. Example sqrt3,sqrt2, π.

#### theorem

1)Theorem: let p be a prime number. If P divides a^2 the p divides a, where a is a positive integer.

Proof: If a^2/p true then a/p is also ture.

a=(p_1p_2...p_n) as per fundamental theorem, this means

a^2= (p_1p_2..p_n)^2

= p_1^2 p_2^2..p_n^2

given: p divides a^2

that means p must belong to pi ,where i lies between 1 to n

Thus, is a factor of a also.

2)Theorem: sqrt2 is irrational

Proof: Let's assume sqrt2 is rational. If sqrt2 is rational the sqrt2= a/b, where a and b are co prime.

sqrt2= a/b

b sqrt2= a

2b^2= a^2 That means 2 divides a^2. And hence 2 is prime number it also divides a.

2×k=a^2 (k is constant)

2×k= a .......eq1

2b^2= 2k^2= 4k^2

b^2= 2k^2, Therefore 2 divides b^2, and since 2 is prime  number it also divides b.

That means 2 is factor of a,b but according to our assumption a and b are co prime which

means they dont have common factors. Here our assumption that sqrt2 is rational is

proven incorrect, thus sqrt is a irrational number.

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Real Numbers part 8 (Irrational Numbers) [00:07:23]
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