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

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 ${\displaystyle \frac{p}{q}}$ where ${\displaystyle p,q\in R}$ and ${\displaystyle q\ne 0}$ 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 ${\displaystyle 12+\sqrt{2}}$ is irrational number.
Yes ${\displaystyle 12\cdot \sqrt{2}=\frac{1}{\sqrt{2}}}$ 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'
${\displaystyle \Rightarrow ap+t=b{p}^{\prime }}$
${\displaystyle \Rightarrow t=b{p}^{\prime }-ap}$
${\displaystyle \Rightarrow t=b{p}^{\prime }+(-ap)}$
${\displaystyle \Rightarrow t=}$ (rational number)+(rational number)
${\displaystyle \Rightarrow t=}$ rational number(sum of two rational numbers is always rational number)
but ${\displaystyle t=}$ 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 $(r\times t)=b{p}^{\prime }$
${\displaystyle \Rightarrow ap\cdot t=b{p}^{\prime }}$
${\displaystyle \Rightarrow t=\frac{b{p}^{\prime }}{ap}}$
${\displaystyle \Rightarrow t=\frac{b{p}^{\prime }}{ap}}$
${\displaystyle \Rightarrow t=}$ (rational number)/(rational number)
${\displaystyle \Rightarrow t=}$ rational number(rational number of two rational numbers is always rational number)
but t= irratational
which is contradiction.
Hence (r*t)= Irrational number. Hence proved.