Finding irrational numbers between two numbers

In this post we will discuss questions related to finding irrational numbers between two given rational numbers.

Before solving the questions, understand two points about irrational numbers;

(a) the decimal value are non terminating and non repeating.
(b) Irrational numbers cannot be expressed in the form of fractions.

Using the above two points, we will solve the below questions.

Question 01
Find the irrational number between \mathtt{0.\overline{2} \ \ \&\ 0.\overline{3}}

Solution
Both the given numbers are rational numbers;

\mathtt{0.\overline{2} \ =\ 0.22222\ .\ \ .\ \ .\ }\\\ \\ \mathtt{0.\overline{3} \ =\ 0.33333\ .\ \ .\ \ .}

Now there can be infinite possible irrational numbers between the above two numbers. One such possible irrational number is given as;

0.250250025000250000 . . . .

Note that the above number is non terminating and non repeating.

Question 02
Find two irrational number between 2/5 and 9/11

Solution
Given below is the decimal value of the fraction;

2/5 = 0.4

9/11 = 0.81818181 . . .

Two irrational numbers between the fractions is given as;

0.5205200520005200005200000 . .
0.6523015189052. . . .

Note that both the above numbers are non terminating and non repeating.

Question 03
Show the example of two irrational numbers whose;

(a) sum is a rational number
(b) sum is irrational number
(c) product is rational number
(d) product is irrational number

Solution
(a) Sum is rational number

Given below A & B are two irrational numbers.

\mathtt{A\ =\ \sqrt{5}}\\\ \\ \mathtt{B=-\sqrt{5}}

Adding the two numbers we will get rational number.

\mathtt{\Longrightarrow \ A\ +\ B}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{5} -\sqrt{5}}\\\ \\ \mathtt{\Longrightarrow \ 0}

Number 0 is a rational number.

(b) Sum is irrational number

Given below A & B are irrational numbers.

\mathtt{A\ =\sqrt{3}}\\\ \\ \mathtt{B=\sqrt{3}}

Adding the two numbers will result in irrational number.

\mathtt{\Longrightarrow \ A\ +\ B}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{3} +\sqrt{3}}\\\ \\ \mathtt{\Longrightarrow \ 2\sqrt{3}}

(c) product is a rational number

Let the two irrational numbers are A & B;

\mathtt{A\ =\sqrt{5} +\sqrt{6}}\\\ \\ \mathtt{B=\sqrt{5} -\sqrt{6}}

Multiplying both the numbers;

\mathtt{\Longrightarrow \ A\ \times \ B}\\\ \\ \mathtt{\Longrightarrow \ \left(\sqrt{5} +\sqrt{6}\right) \ \left(\sqrt{5} -\sqrt{6}\right)}\\\ \\ \mathtt{\Longrightarrow \ \left(\sqrt{5}\right)^{2} -\left(\sqrt{6}\right)^{2}}\\\ \\ \mathtt{\Longrightarrow \ 5\ -\ 6}\\\ \\ \mathtt{\Longrightarrow \ -1}

Hence, on multiplication we get a rational number.

(d) Product is irrational number.

Let the two numbers are A & B.

\mathtt{A\ =\sqrt{2}}\\\ \\ \mathtt{B=\sqrt{3}}

Multiplying both the numbers.

\mathtt{\Longrightarrow \ A\ \times \ B}\\\ \\ \mathtt{\Longrightarrow \sqrt{2} \times \sqrt{3}}\\\ \\ \mathtt{\Longrightarrow \ \sqrt{6}}

Hence, on multiplication we get irrational number.

Question 04
Which of the below irrational number lies between 2 and 2.5

\mathtt{( a) \ \sqrt{7}}\\ \\ \mathtt{( b)\sqrt{5}}\\ \\ \mathtt{( c)\sqrt{3}}\\ \\ \mathtt{( d) \ \sqrt{11}}

Solution
The approximate value of all the irrational numbers is given below;

\mathtt{( a) \ \sqrt{7} \ =\ 2.645}\\ \\ \mathtt{( b)\sqrt{5} \ =\ 2.236}\\ \\ \mathtt{( c)\sqrt{3} =\ 1.73}\\ \\ \mathtt{( d) \ \sqrt{11} =3.16}

Hence, option (b) is correct.

Question 05
Find three irrational number between 2 and 3.

Solution
Three irrational numbers are given below;

2.10100100010000100000 . . . .
2.53681634917462231 . . .
2.717117111711117111111 . . .

Note that all numbers lie between 2 & 3 and they are non repeating, non terminating decimals.

Next chapter : Draw irrational number graphically



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