Question

prove that
$\sqrt{5}$
is irrational

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Answer 1

${\huge {\bold{\underline{\green {Answer:}}}}}$

Given:

$\sqrt{5}$

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Asked:

To prove that $\sqrt{5}$ is irrational.

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Solution:

Let us assume√5 is a rational number.

p/q= √5.

p =√5q

squaring both sides

p^2=5q^2..... -1

= 5 divides p^2

= 5 divides p.... -2

p =5m(where m is an integer).. -3

put 3 in 1

(5m) ^2= 5q^2

25m^2=5q^2

5m^2 =q^2

= 5 divides q^2

= 5 divides q

5=q...-4

from 2 & 4

we arrive at a contradiction because aur assumption that√5is a rational number is wrong

so,

√5 is an irrational number

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Answer 2

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$\huge\tt{TO~PROOF:}$

$\sqrt{5}$ is irrational

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$\huge\tt{PROVING:}$

Let us assume that √5 is a rational number

➩a/b = √5

➩a = √5b

If we square both the numbers,

➩a² = 5b² ___________(EQ.1)

➩5 divides a²

➩5 divides p ___________ (EQ.2)

➩a = 5 m m is an integer _________(EQ.3)

putting EQ.3 in EQ.1

➩5m² = 5b²

➩25m² = 5b²

➩5m² = b²

➩5 divides b²

➩5 divides b

➩5 = b ________(EQ.4)

now, from EQ.2&4

we can say that √5 isn't a rational number

so, it's an irrational number

Hence, proven that √5 is an irrational number

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