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

EunoR 2021-03-09 Answered
Discover prove: Combining Rational and Irrationalnumbers is 1.2+2 rational or irrational? Is 122 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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Isma Jimenez
Answered 2021-03-10 Author has 84 answers

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 pq where p,qR and q0 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+2 is irrational number.
Yes 122=12 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=bp
t=bpap
t=bp+(ap)
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 (r×t)=bp
apt=bp
t=bpap
t=bpap
t= (rational number)/(rational number)
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.

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