"Prove that2 is an irrational number." solved and explained

Harmony Oneal

Answered question

2022-11-24

Prove that

2

is an irrational number.

Kendrick Lamb

Beginner2022-11-25Added 6 answers

Proof of

2

is an irrational numbers.
Assume,

2

is a rational number, and its representation is

\frac{p}{q}

, in which

p

and

q

are co-prime integers and

q \neq 0

,
i.e.

2 = \frac{p}{q}

. where,

p

and

q

are coprime numbers, and

q \neq 0

.
On squaring both sides of the above equation;

{(2)}^{2} = {(\frac{p}{q})}^{2} \Rightarrow 2 = \frac{p^{2}}{q^{2}} \Rightarrow 2 q^{2} = p^{2} . . . (i) \Rightarrow p^{2} is a multiple of 2 \Rightarrow p is a multiple of 2 . . . (i i)

Since,

p

is a multiple of two.

∴ p = 2 m \Rightarrow p ² = 4 m ² \dots (i i i)

Using equation

(i)

into the equation

(i i i)

, we get;

2 q ² = 4 m ² \Rightarrow q ² = 2 m ² \Rightarrow q^{2} is a multiple of 2 \Rightarrow q is a multiple of 2 . . . (i v)

Equation

(i i) and (iv)

, implies that

p

and

q

have a common factor of 2 in mathematics. They are not co-primes, which goes against our incorrect premise that

2

is a rational number.
Hence,

2

is an irrational number(proved)

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