Real number between Two Unequal Real Numbers

We will learn here ‘how to find a real number between two unequal real numbers?’.

If x, y are two real numbers,$$\frac{x + y}{2}$$ is a real number lying between x and y.

If x, y are two positive real numbers, $$\sqrt{xy}$$ 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 a rational number, $$\sqrt{xy}$$ is an irrational number lying between x and y,

Solved examples to find real numbers between two real numbers:

1. Insert two irrational numbers between √2 and √7.

Solution:

Consider the squares of √2 and √7.

$$\left ( \sqrt{2} \right )^{2}$$ =2 and $$\left ( \sqrt{7} \right )^{2}$$ = 7.

Since the numbers 3 and 5 lie between 2 and 7 i.e., between $$\left ( \sqrt{2} \right )^{2}$$ and $$\left ( \sqrt{7} \right )^{2}$$, therefore, √3 and √5 lie between √2 and √7.

Hence two irrational numbers between √2 and √7 are √3 and √5.

Note: Since infinitely many irrational numbers between two distinct irrational numbers, √3 and √5 are not only irrational numbers between √2 and √7.

2. Find an irrational number between √2 and 2.

Solution:

A real number between √2 and 2 is $$\frac{\sqrt{2} + 2}{2}$$, i.e., 1 + $$\frac{1}{2}$$√2.

But 1 is a rational number and $$\frac{1}{2}$$√2 is an irrational number. As the sum of a rational number and an irrational number is irrational, 1 + $$\frac{1}{2}$$√2 is an irrational number between √2 and 2.

3. Find an irrational number between 3 and 5.

Solution:

3 × 5 = 15, which is not a perfect square.

Therefore, $$\sqrt{15}$$ is an irrational number between 3 and 5.

4. Write a rational number between √2 and √3.

Solution:

Take a number between 2 and 3, which is a perfect square of a rational number. Clearly 2.25, i.e., is such a number.

Therefore, 2 < (1.5)$$^{2}$$ < 3.

Hence,√2 < 1.5 √3.

Therefore, 1.5 is a rational number between √2 and √3.

Note: 2.56, 2.89 are also perfect squares of rational numbers lying between 2 and 3. So, 1.67 and 1.7 are also rational numbers lying between √2 and √3.

There are many more rational numbers between √2 and √3.

5. Insert three rational numbers 3√2 and 2√3.

Solution:

Here 3√2 = √9 × √2 = $$\sqrt{18}$$  and 2√3 = √4 × √3 = $$\sqrt{12}$$.

13, 14, 15, 16 and 17 lies between 12 and 18.

Therefore, $$\sqrt{13}$$, $$\sqrt{14}$$, $$\sqrt{15}$$ and $$\sqrt{17}$$ are all the rational numbers between 3√2 and 2√3.

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