# Discover prove: Combining Rational and Irrationalnumbers is 1.2+sqrt2 rational or irrational? Is 1/2*sqrt2 rational or irrational? Experiment with sum

Discover prove: Combining Rational and Irrationalnumbers is $1.2+\sqrt{2}$ rational or irrational? Is $\frac{1}{2}\cdot \sqrt{2}$ rational or irrational? Experiment with sums and products of ther rational and irrational numbers. Prove the followinf.
(a) The sum of rational number r and an irrational number t is irrational.
(b) The product of a rational number r and an irrational number t is irrational.
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Solution:- Combining Rational numbers and irrational numbers:
Rational Numbers:- Numbers which either terminating or none-terminating Repeating. and can be convertible in the form of $\frac{p}{q}$ where $p,q\in R$ and $q\ne 0$ is known as Rational numbers
Irrational numbers:- The Numbers which are non- terminating and non-repeating and all square root of prime numbers is known as Irrational numbers.
Yes $12+\sqrt{2}$ is irrational number.
Yes $12\cdot \sqrt{2}=\frac{1}{\sqrt{2}}$ is irrational number.
a) prove that sum of rational number and irratoional number is irrational.
Given that :− r be a rational number and t be an irrational number.To prove:− sum of rational number and irratoional number is irrational.
i.e. (r+t)=irrational
Proof:−It is given that r is rational, t is irrational,and assume that is (r+t) rational.
Since a and a+b are rational, we can write them as fraction. Let, r=ap and (r+t)=bp'
$⇒ap+t=b{p}^{\prime }$
$⇒t=b{p}^{\prime }-ap$
$⇒t=b{p}^{\prime }+\left(-ap\right)$
$⇒t=$ (rational number)+(rational number)
$⇒t=$ rational number(sum of two rational numbers is always rational number)
but $t=$ irratational which is contradiction.
Hence (r+t)= Irrational number.
Hence proved.
b) prove that product of rational number and irratoional number is irrational.
Given that :− r be a rational number and t be an irrational number.
To prove:−product of rational number and irratoional number is irrational.
i.e. (r*t)=irrational
Proof:−It is given that r is rational, t is irrational, and assume that is (r*t) rational.
Since a and a*b are rational, we can write them as fraction
r=ap
and $\left(r×t\right)=b{p}^{\prime }$
$⇒ap\cdot t=b{p}^{\prime }$
$⇒t=\frac{b{p}^{\prime }}{ap}$
$⇒t=\frac{b{p}^{\prime }}{ap}$
$⇒t=$ (rational number)/(rational number)
$⇒t=$ rational number(rational number of two rational numbers is always rational number)
but t= irratational
Hence (r*t)= Irrational number. Hence proved.