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Question Number 158567 by MathsFan last updated on 06/Nov/21

show that (√3) is an irrarional number

$$\:\mathrm{show}\:\mathrm{that}\:\sqrt{\mathrm{3}}\:\mathrm{is}\:\mathrm{an}\: \\ $$$$\:\mathrm{irrarional}\:\mathrm{number} \\ $$

Commented by mr W last updated on 06/Nov/21

assume (√3)=(p/q) with gcd(p,q)=1 p^2 =3q^2 ⇒p=3k q^2 =3k^2 ⇒q=3h ⇒gcd(p,q)=3 ⇒contradition with gcd(p,q)=1 ⇒(√3)≠(p/q)

$${assume}\:\sqrt{\mathrm{3}}=\frac{{p}}{{q}}\:{with}\:{gcd}\left({p},{q}\right)=\mathrm{1} \\ $$$${p}^{\mathrm{2}} =\mathrm{3}{q}^{\mathrm{2}} \\ $$$$\Rightarrow{p}=\mathrm{3}{k} \\ $$$${q}^{\mathrm{2}} =\mathrm{3}{k}^{\mathrm{2}} \\ $$$$\Rightarrow{q}=\mathrm{3}{h} \\ $$$$\Rightarrow{gcd}\left({p},{q}\right)=\mathrm{3}\: \\ $$$$\Rightarrow{contradition}\:{with}\:{gcd}\left({p},{q}\right)=\mathrm{1} \\ $$$$\Rightarrow\sqrt{\mathrm{3}}\neq\frac{{p}}{{q}} \\ $$

Commented by MathsFan last updated on 06/Nov/21

thank you sir

$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$

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