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**

- Decimal Representation of Irrational Number
- Comparison between Two Irrational Numbers
- Comparison between Rational and Irrational Numbers
- Definition of Irrational Numbers

### 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.

### Frequently Asked Question’s

**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.