RATIONAL NUMBERS - 4

RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS.

Between any two real numbers, there are countable numbers, and sometimes, there are no number at all. For example, between 12 and 20 there are seven integers (13, 14, 15, 16, 17, 18, 19), and between 10 and 11 there are no integers. Between 100 and 1000 there are 899 numbers. However, we can not observe this limitation in rational numbers. In fact, between any two rational numbers, we can find countless rational numbers. Check this fact through the following example.

FIRST METHOD:

Example: How many rational numbers are there between 1/5 and 4/5?

Solution: You may say there are only two – 2/5, 3/5

But if you rewrite 1/5 as 5/25 and 4/5 as 20/25, then you would have

6/25, 7/25, 8/25, 9/25, 10/25………..10/25.

In all, there are 14 rational numbers and all of them lie between 1/5 and 4/5.

Now if we write 1/5 as 100/500 and 4/5 as 400 500, we would have a new set of rational numbers

occurring between 1/5 and 4/5.

These are 101/500, 102/500, 103/500, 104/500, 105/500,…………..397/500, 398/500, 399/500.

And, you can go on endlessly finding new sets of rational numbers between 1/5 and 4/5/

This method of finding rational numbers can be applied to any two randomly selected rational numbers.

Therefore, it follows that there are countless rational numbers between any two rational numbers.

  

Second method: This method is also called the arithmetic mean method. In this method, first we find the mean of the two given rational numbers, then we go on finding more mean values of either the new and original rational number combination or to the tow mean values.

Suppose two given rational numbers are a and b. We will find rational numbers q1, q2, q3, q4…..between

a nd b as follows.

                                    q1 = 1/2 (a + b)

                                    q2 = 1/2 (a + q1)

                                    q3 = 1/2 (q1 + b)

                                    q4 = 1/2 (q1 + q2)

Counting this procedure, we can get the required number of rational numbers between the rational

numbers and b . Observe the following example that is based on this method.

Example: Find four ration numbers between 1 and 2, by method of arithmetic mean.

Solution:          q1 = 1/2 (1 +2) = 1/2 × 3 = 3/2

                        q2 = 1/2 (1 + 3/2) = 1/2 × (2 + 3/2) = 5/4

                        q3 = 1/2 (3/2 + 2) = 1/2 × ( 3 + 4/2) = 7/4

                        q4 = 1/2 (3/2 + 5/4) = ½ (6 +5/4) = 11/8

If we represent these numbers on the number line, you will notice that all these numbers lies between 1

and 2.

                                                                                5/4       3/2                                                             

<----------------------|---|----|----|---|----|----|---|---|---|---|---|---|-------à

                                                             1          11/8      7/4     2

   

Example: Find a rational number lying between 1/3 and 1/2

Solution: Let x = 1/3 and y = 1/2.

Then, clearly x < y

Then, a rational number lying between the given numbers

                  = 1/2(x + y)

                  = 1/2(1/3 + 1/2)

                  = (1/2 × 5/6)

                  = 5/12

Example 3: Insert five rational numbers between 3/5 and 2/3.

Solution:     Suppose that x = 3/5 and y = 2/3

               

Then, 3/5 < 2/3, since ( 3 × 3) < (5 × 2).

                  So, x < y,

                  Where x = 3/5 and y = 2/3

                  (y – x) = (2/3 – 2/5) = 1/15

Hence         n = 5 and therefore, (n + 1) = 5 + 1 = 6

Hence        (y – x)/(n +1) = ( 1/15 × 1/6) = 1/90

So the five rational numbers between 3/5 and 2/3 are

(3/5 + 1/90), (3/5 + 2/90), (3/5 + 3/90), (3/5 + 4/90) and (3/5 + 5/90)

So the required numbers are 55/90, 56/90, 57/90, 58/90 and 59/90.

Remark: While finding any number of rational numbers lying between two decimals, we may take the value of n larger than the required number, as per our convenience, making (y – x)/(n – 1) a terminating decimal. Then, we write only the required number of rational numbers and leave the others aside.

Example: Find the 20 rational numbers between 0 and 0.1.

Solution: For convenience we take n = 24 so that (n + 1) = 25.

Here x = 0 and y = 0.1 and therefore , (y – x) = 0.1

So (y – x)/(n + 1) = 0.1/25 = 0.004

Twenty rational numbers lying between 0 and 0.1 are:

(0 + 0.004),      (0 + 2 × 0.004),                        (0 + 3 × 0.004),                        (0 + 4 × 0.004),                        (0 + 5 × 0.004),

            (0 + 6 × 0.004)………………..……………………………………………………(0 + 20 ×0.004)

Hence, the required numbers are:

  (0.004, 0.008, 0.012, 0.016, 0.020, 0.024, 0.028, 0.032, 0.036, …………………0.072)