Tag: properties of irrational numbers

Irrational numbers have decimal expansions that neither terminate nor repeating

In the given options, 

The value $\dfrac{22}7$ is a rational number, as it can be expressed in the form $\dfrac pq$. 

An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
$\sqrt5$ is irrational as it can never be expressed in the form a/b

An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
$7\sqrt5$ is irrational as it can never be expressed in the form a/b

An irrational number is any real number that cannot be expressed as a ratio a/b, where a and b are integers and b is non-zero.
$1/(\sqrt2)$ is irrational as it can never be expressed in the form a/b

Let's assume that $3+2\sqrt5$ is rational..... 

then 

$3+2\sqrt5 = p/q $

$\sqrt5 =( p-3q)/(2q) $ 

now take $p-3q$ to be P and $2q$ to be Q........where P and Q are integers 

which means, $\sqrt5= P/Q$...... 

But this contradicts the fact that $\sqrt5$ is rational 

So our assumption is wrong and $3+2\sqrt5$ is irrational.

A rational number is any number that can be expressed as a fraction $\dfrac pq$ of two integers with $q$ not equal to zero.
As in the case of $\sqrt2$ and $\sqrt3$, it cannot be expressed as a fraction $\dfrac pq$.

The basic definition for a rational number is that it can be represented in the form of $p/q$, where p and q are integers and q is a non-zero integer. Here, $\sqrt7$ is not a perfect square and thus cannot be expressed in the form of $p/q$, thus it is an irrational number.

A surd, by its very definition is an irrational number.