 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.

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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.