Question

Show that √2 is an irrational number and hence prove that 1 + √2 is also an irrational number​

Answer 1

Need to ProvE :-

• √2 is an irrational Number .
• 1 + √2 is a Irrational Number .

$\red{\frak{Given}}\Bigg\{ \sf A \ number \ \sqrt2.$

Given number to us is √2 . On the contrary let us assume that √2 is a Rational number .So it can be expressed in the form of p/q where p and q are integers and q is not equal to zero. Also p and q are co - primes .

Therefore ,

$\sf\longrightarrow \sqrt2 =\dfrac{p}{q}$

Square both sides ,

$\sf\longrightarrow (\sqrt2 )^2=\bigg\{\dfrac{p}{q} \bigg\}^2 \\\\\\\sf\longrightarrow 2 =\dfrac{p^2}{q^2} \\\\\\\sf\longrightarrow 2q^2 = p^2 \dots\dots (i)$

• This implies that , 2 is a factor of p² . Therefore by fundamental theorem of arithmetic we can say that 2 divides p also.

Let ,

$\sf\longrightarrow p = 2k$

Put this value in (i) ,

$\sf\longrightarrow 2q^2 = (2k)^2 \\\\\\\sf\longrightarrow 2q^2 = 4k^2 \\\\\\\sf\longrightarrow q^2 = 2k^2$

• This implies that , 2 is a factor of q² . Therefore by fundamental theorem of arithmetic we can say that 2 divides q also.

This contradicts our assumption that p and q are coprime since we found 2 as their factor . Therefore our assumption was wrong .

$\bf\longrightarrow \sqrt2 = Irrational \ Number$

• Now we know that , the sum of a rational number and a Irrational Number is Irrational . Therefore the sum of 1 and √2 will be Irrational . As 1 is a Rational number and √2 is a Irrational Number.

$\bf\longrightarrow \sqrt2 + 1 = Irrational \ Number$

Hence Proved !

Answer 2

Step-by-step explanation:

i hope it will help you....

both are irrational numbers