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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