Tag: rational numbers

Quizzes (14) Questions (65)

Questions Related to rational numbers

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers
State whether the given statement is True or False :

The number $6+\sqrt { 2 } $ is irrational.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$Here \  6 \text{ is a  rational number and }$$\sqrt2$ is a $irrational$ number


And $\text{the addition of rational and irrational is always an irrational number}$

So that $(6+\sqrt2)$ is an irrational number.

hence option A is correct

Multiple choice maths rational numbers proof of irrationality of numbers proofs of irrationality introduction to irrational numbers
State whether the given statement is True or False :
If $p,  q $ are prime positive integers, then $\sqrt { p } +\sqrt { q } $ is an irrational number.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
$\sqrt{p}+\sqrt{q}$ is rational ........ assumption
$\sqrt{p}+\sqrt{q}=\dfrac{a}{b}$
squaring, we get
$p+q+2\sqrt{pq}=\left(\dfrac{a}{b}\right)^{2}$

$\sqrt{pq}=\dfrac{1}{2}\left[\left(\dfrac{a}{b}\right)^{2}-p-q\right] - (i)$

Now, $p$ & $q$ are prime positive numbers so, $\sqrt{p}$ and $\sqrt{q}$ is irrational also $\sqrt{pq}$ 

so in (i)
Irrational $=$ rational $\Rightarrow$ which is a contradiction

$\Rightarrow\ \sqrt p+\sqrt q$ is a irrational number if $p,q$ are prime positive numbers