Name: Rationalisation of Surds
Uploaded: 2019-07-19
Duration: 5 min 28 s
Description: When the denominator of an expression is a surd that can be reduced to an expression with a rational denominator, this process is called the rationalisation of a surd.

Question

Find six rational numbers between 3 and 4.

Accepted Answer

Given,

To write the rational numbers between $3 and 4$ .

We know,

If $a, b$ are any two rational numbers, then $\frac{a + b}{2}$ is a rational number and it lies between them.

i.e., If $a < b$ , then $a < \frac{a + b}{2} < b$ .

Therefore, we get the rational numbers between $3, 4$ :

A rational number between $3 and 4 =$ $\frac{3 + 4}{2} = \frac{7}{2}$

A rational number between $3 and \frac{7}{2} =$ $\frac{3 + \frac{7}{2}}{2} = \frac{\frac{13}{2}}{2} = \frac{13}{4}$

A rational number between $3 and \frac{13}{4} =$ $\frac{3 + \frac{13}{4}}{2} = \frac{\frac{25}{4}}{2} = \frac{25}{8}$

A rational number between $\frac{7}{2} and 4 =$ $\frac{\frac{7}{2} + 4}{2} = \frac{\frac{15}{2}}{2} = \frac{15}{4}$

A rational number between $\frac{15}{4} and 4 =$ $\frac{\frac{15}{4} + 4}{2} = \frac{\frac{31}{4}}{2} = \frac{31}{8}$

A rational number between $\frac{31}{8} and 4 =$ $\frac{\frac{31}{8} + 4}{2} = \frac{\frac{63}{8}}{2} = \frac{63}{16}$

Thus, we get the rational numbers as $3, \frac{7}{2}, \frac{13}{4}, \frac{25}{8}, \frac{15}{4}, \frac{31}{8}, \frac{63}{16} and 4$ .

Hence, three rational numbers between $3 and 4$ are $\frac{7}{2}, \frac{15}{4}, \frac{13}{4}, \frac{31}{8}, \frac{25}{8}, \frac{63}{16}$ .

Find six rational numbers between 3 and 4.

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