In simple words, real numbers are the combination of both rational and irrational numbers in the number system. Irrational numbers can never be written in $$\frac { a }{ b }$$ form. The decimal expansion of irrational numbers is neither terminating nor recurring. And rational numbers are the opposite of irrational numbers. Here, we will teach how to find a real number between two unequal real numbers and solved questions.

## Real Numbers between Two Unequal Real Numbers

In general, both rational and irrational numbers are real numbers. Rational numbers can be expressed as a simple fraction and they are terminating numbers in decimal form. Irrational numbers are the square root, cube root numbers and those are non-terminating. Below mentioned are the formulas to get the real numbers between any two unequal real numbers.

• If x, y are two real numbers, then $$\frac { x+y }{ 2 }$$ is a real number that lies between x and y.
• If x, y are two positive real numbers, then √(xy) is a real number lying between x and y.
• If x, y are two positive real numbers such that x * y is not a perfect square of rational number, then √(xy) is an irrational number that lies between x and y.

Also, find

### Examples to Find Rational & Irrational Numbers Between Real Numbers

Example 1:
Find an irrational number between √5 and 5.

Solution:
Given two real numbers are √5 and 5
A real number between √5 and 5 is $$\frac { √5+5 }{ 2 }$$ = ½√5 + 1
But 1 is a rational number and ½√5 is an irrational number. The sum of a rational number and an irrational number is irrational.
So, ½√5 + 1 is an irrational number that lies between √5 and 5.

Example 2:
Find a rational number between √5 and √7.

Solution:
Take a number between 5 and 7, which is a perfect square of a rational number. Clearly, 6.25 is such a number
Therefore, 5 < (2.5)² < 7
Hence, √5 < 2.5 < √7
Therefore, 2.5 is a rational number that lies between √5 and √7.

Example 3:
Insert two irrational numbers between √3 and √11.

Solution:
Given two real numbers are √3 and √11
Consider the squares of √3, √11
(√3)² = 3
(√11)² = 11
Since the numbers 5, 7 lie between 3 and 11 i.e between (√3)² and (√11)²
Therefore, √3 and √11 are √7 and √5.
Hence two irrational numbers between √3 and √11 are √5 and √7.

Example 4:
Insert irrational numbers between 3√2 and 2√3.

Solution:
Here, 3√2 = √9 x √2 = √18
2√3 = √4 x √3 = √12
13, 14, 15, 16, 17 lies between 12 and 18.
Therefore, √13, √4, √15 and √17 are all irrational numbers between 3√2 and 2√3.

1. How many rational numbers between two unequal rational numbers?

There is a possibility of infinite number of rational numbers lies between two unequal rational numbers.

2. Is there a real number between any two real numbers?

Yes, there is a real number between any given two real numbers.

3. How many rational numbers lies between 2 and 3?

There are six six rational numbers lies between real numbers 2 and 3. They are 2.1, 2.2, 2.3, 2.4, 2.5, and 2.6.

4. How to find real number between two unequal real numbers?

If a, b are two unequal real numbers, then $$\frac { a+b }{ 2 }$$ is a real number that lies between a and b.