More about rationals — lesson. Mathematics State Board, Class 9.

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Methods to find the rational number between any two rational numbers:

Method 1: [Average method]

Let the rational number be $a$ and $b$ .
Add $a$ and $b$ and divide the sum by $2$ . That is, $(a+b)/2$ which might lie in between those two number.
To get another rational number, find the average of $c$ and $a$ . For one more rational number, find the average of $c$ and $b$ . In this way, you can find the infinite number of irrational between two rational numbers.

Method 2: [Same denominator method]

This method gives all the required number of rational numbers between

$a$ and

$b$ in one step.

If we want to find

$n$ rational numbers between the numbers

$a$ and

$b$ , we write

$a$ and

$b$ as rational numbers with denominator

$n + 1$ . That is, make it as

$a = \frac{a \times (n + 1)}{n + 1}, b = \frac{b \times (n + 1)}{n + 1}$ . Now we can verify that the numbers between

$\frac{a \times (n + 1)}{n + 1}$ and

$\frac{b \times (n + 1)}{n + 1}$ are all rational numbers between

$a$ and

$b$ .

Example:

Let the rational numbers be

$6$ and

$7$ . Now follow the steps to find the rational numbers.

Add the rational numbers

$6$ and

$7$ and divide the sum by

$2$ .

That is

$\frac{6 + 7}{2} = \frac{13}{2}$ .

Now let's use the second method to find the set of four rational between the numbers

$6$ and

$7$ .

Here

$a = 6, b = 7$ and

$n = 4$ .

Substituting the known values, we will have

$\frac{6 \times (4 + 1)}{4 + 1} = \frac{30}{5}$ and

$\frac{7 \times (4 + 1)}{4 + 1} = \frac{35}{5}$ .

Thus the number between

$30/5$ and

$35/5$ are

$31/5, 32/5, 33/5$ and

$34/5$ .

Therefore, the four rational numbers are

$31/5, 32/5, 33/5$ and

$34/5$ .

Important!

In the same way, we can find as many rational numbers between two rational numbers. Thus, there are infinitely many rational numbers between any two given rational numbers.

Rational number

$Q$ does not have a unique representation in the form of

$p/q$ .

$\frac{3}{4} = \frac{6}{8} = \frac{9}{12} = \frac{12}{16}$ and so on. These are called an equivalent rational number.

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3. More about rationals

Theory: