Answer

Verified

428.1k+ views

**Hint:**

Here, we have to prove that the given number is an irrational number. An irrational number is a number which cannot be expressed as a fraction or the ratio of two integers. An irrational number can neither be terminating or recurring.

**Complete step by step solution:**

We have to prove that $\left( 5-2\sqrt{3} \right)$ is an irrational number.

To prove that the number $\left( 5-2\sqrt{3} \right)$ is an irrational number, we have to prove that the number $5-2\sqrt{3}$is not a rational number.

A rational number is a number which can be expressed as a fraction or the ratio of two integers as $\dfrac{p}{q}$.

Now, considering $5-2\sqrt{3}$ as a rational number.

Expressing the number as rational number, we get

\[ \Rightarrow 5 - 2\sqrt 3 = \dfrac{p}{q}\]

Rewriting the equation, we get

\[ \Rightarrow - 2\sqrt 3 = \dfrac{p}{q} - 5\]

By taking L.C.M, and adding the like terms, we get

\[ \Rightarrow - 2\sqrt 3 = \dfrac{p}{q} - 5 \times \dfrac{q}{q}\]

$\Rightarrow -2\sqrt{3}=\dfrac{p}{q}-\dfrac{5q}{q}$

$\Rightarrow -2\sqrt{3}=\dfrac{p-5q}{q}$

Rewriting the equation, we get

\[ \Rightarrow \sqrt 3 = \dfrac{{p - 5q}}{{ - 2q}}\]

We know that the surd is always an irrational number.

So, $\sqrt{3}$ is an irrational number whereas $\dfrac{p-5q}{-2q}$ is a rational number.

So, we have $L.H.S\ne R.H.S$.

So, our assumption that $5-2\sqrt{3}$ is a rational number is wrong.

Hence, $5-2\sqrt{3}$ is an irrational number.

Therefore, $5-2\sqrt{3}$ is an irrational number.

Hence, proved.

**Note:**

We should know the properties of irrational numbers such that addition of two irrational numbers may or may not be irrational. Subtraction of two irrational numbers may or may not be irrational. But the difference between a rational number and an irrational number is always irrational.

We know that surd is defined as the number which cannot be simplified to find the square root. Every rational number is not a surd but every irrational number is a surd. Here, we have used the concept of contradiction. Contradiction is a method of proving the statement true by showing the assumption to be false. This method is quite easy to prove that $5-2\sqrt{3}$ is an irrational number.

Recently Updated Pages

what is the correct chronological order of the following class 10 social science CBSE

Which of the following was not the actual cause for class 10 social science CBSE

Which of the following statements is not correct A class 10 social science CBSE

Which of the following leaders was not present in the class 10 social science CBSE

Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE

Which one of the following places is not covered by class 10 social science CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How do you graph the function fx 4x class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference Between Plant Cell and Animal Cell

Why is there a time difference of about 5 hours between class 10 social science CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Draw a labelled sketch of the human eye class 12 physics CBSE