Root 2 is irrational (anglický)
Hľadať " Root 2 is irrational " pomocou NRICH | PLUS | maths.org | Google
Definícia vek 16 úroveň 3
√2 is irrational.
Justification: If √2 were rational we could write √2 = [(p)/(q)] with p, q integers having no common factors. But then we have 2 = [(p2)/(q2)] by squaring both sides, and 2q2=p2.
The left hand side has a factor of 2, so the right hand side must also have a factor of 2, so p must be divisible by 2, i.e. we can write p=2r. Then we have 2q2=4r2 or q2=2r2. This means that q must have a factor of 2. So p and q have a common factor of 2, which contradicts our assumption that p and q have no common factors. So we conclude that we cannot write √2 as a fraction, so it is irrational.
Justification: If √2 were rational we could write √2 = [(p)/(q)] with p, q integers having no common factors. But then we have 2 = [(p2)/(q2)] by squaring both sides, and 2q2=p2.
The left hand side has a factor of 2, so the right hand side must also have a factor of 2, so p must be divisible by 2, i.e. we can write p=2r. Then we have 2q2=4r2 or q2=2r2. This means that q must have a factor of 2. So p and q have a common factor of 2, which contradicts our assumption that p and q have no common factors. So we conclude that we cannot write √2 as a fraction, so it is irrational.
Prepojenia
- širší:
- (en) The square root of any prime number is irrational
- (en) Theorem
- odkazy na iné termíny:
- (en) Irrational number
- (en) Square root
Finančná podpora: