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

Irrational numbers
ANSWERED Discover prove: Combining Rational and Irrationalnumbers is $$\displaystyle{1.2}+\sqrt{{2}}$$ rational or irrational? Is $$\displaystyle\frac{{1}}{{2}}\cdot\sqrt{{2}}$$ 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. 2021-03-10
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'
$$\displaystyle\Rightarrow{a}{p}+{t}={b}{p}'$$
$$\displaystyle\Rightarrow{t}={b}{p}'−{a}{p}$$
$$\displaystyle\Rightarrow{t}={b}{p}'+{\left(−{a}{p}\right)}$$
$$\displaystyle\Rightarrow{t}=$$ (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 PSK(r×t)=bp'
$$\displaystyle\Rightarrow{a}{p}\cdot{t}={b}{p}'$$
$$\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}$$
$$\displaystyle\Rightarrow{t}=\frac{{{b}{p}'}}{{{a}{p}}}$$
$$\displaystyle\Rightarrow{t}=$$ (rational number)/(rational number)
$$\displaystyle\Rightarrow{t}=$$ rational number(rational number of two rational numbers is always rational number)
but t= irratational