Question

Show root √2 is an irrational number..

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Answer 1

$\Large{\underline{\underline{\bf{Answer:-}}}}$

Let us assume that,

√2 is a rational number of simplest form $\frac{a}{b}$, having no common factor other than 1.

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  √2 = $\frac{a}{b}$

On squaring both sides, we get ;

        2 = $\frac{a^{2}}{b^{2}}$

   ❥ a² = 2b²

Clearly, a² is divisible by 2.

So, a is also divisible by 2.

Now, let some integer be c.

❥ a = 2c

Substituting for a, we get ;

❥ 2b² = 2c

Squaring both sides,

❥ 2b² = 4c²

❥ b² = 2c²

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❧ Thís mєαns thαt, 2 dívídєs в², αnd sσ 2 dívídєs в.

thєrєfσrє, α αnd в hαvє αt lєαst 2 αs α cσmmσn fαctσr. thís cσntrαdícts .

Thís cσntrαdíctíσn hαs αrísєs вєcαusє σf σur αssumptíσn thαt √2 ís rαtíσnαl.

So, √2 is irrational.

$\Large{\underline{\underline{\bf{ADDITIONAL Information:-}}}}$

ྉ What is Rational Number ?

❥ A rαtíσnαl numвєr ís α numвєr thαt cαn вє єхprєssєd αs thє quσtíєnt σr frαctíσn p/q σf twσ íntєgєrs, α numєrαtσr p αnd α nσn-zєrσ dєnσmínαtσr q. síncє q mαч вє єquαl tσ 1, єvєrч íntєgєr ís α rαtíσnαl numвєr.

ྉ What is Irrational Number ?

❥ Thє írrαtíσnαl numвєrs αrє αll thє rєαl numвєrs whích αrє nσt rαtíσnαl numвєrs. thαt ís, írrαtíσnαl numвєrs cαnnσt вє єхprєssєd αs thє rαtíσ σf twσ íntєgєrs. ... írrαtíσnαl numвєrs cαn αlsσ вє єхprєssєd αs nσn-tєrmínαtíng cσntínuєd frαctíσns αnd mαnч σthєr wαчs.

ྉ How to prove a number be Irrational ?

❥ Thє usuαl αpprσαch ís “prσσf вч cσntrαdíctíσn” – σnє σf thє mσst pσwєrful αnd usєful prσσf tєchníquєs ín mαthєmαtícs. чσu stαrt вч αssumíng thαt α numвєr ís rαtíσnαl, αnd thєn shσw thαt thís lєαds tσ α lσgícαl cσntrαdíctíσn. thís dєmσnstrαtєs thαt чσur ínítíαl αssumptíσn must вє fαlsє, sσ thє numвєr must вє írrαtíσnαl.