Expert Answer to "Prove: there is a positive irrational number x..."

Jeffery Clements

Answered question

2022-06-20

Prove: there is a positive irrational number

x

for which

x^{2} = 3

lorienoldf7

Beginner2022-06-21Added 19 answers

The range of the function

f (x) = x^{2}

R^{+}

R^{+}

(this is proved with the intermediate value theorem). So there exists a positive real

α

with

α^{2} = 3

.
Suppose a rational exists, of the form

\frac{p}{q}

with

p

and

q

coprime such that

(\frac{p}{q})^{2} = 3

, we have that

p^{2} = 3 q^{2}

, it follows that

p

is a multiple of

3

(by euclids theorem, since

3

is prime). So let

p = 3 r

, notice that we have

3 r^{3} = q^{2}

, so

3

also divides

q

, a contradiction.

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