RATIONAL NUMBERS(LECTURE 7)

Rational Numbers between Two Rational Numbers

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Rational Numbers between Two Rational Numbers

• Can you tell the natural numbers between 1 and 5? They are 2, 3 and 4.
• How many natural numbers are there between 7 and 9? There is one and it is 8.
• How many natural numbers are there between 10 and 11? Obviously none.
• List the integers that lie between –5 and 4. They are – 4, – 3, –2, –1, 0, 1, 2, 3.
• How many integers are there between –1 and 1?
• How many integers are there between –9 and –10?

You will find a definite number of natural numbers (integers) between two natural
numbers (integers).

EXPECTED LEARNING OUTCOME

• Today, we will be able to find how many rational numbers are there between two rational numbers.

CLICK THE LINK BELOW TO  UNDERSTAND BETTER!!!!!!!!!

https://youtu.be/kUsOhK9YCu8

Hope now you are comfortable with this concept........

 An important result: If a/b and c/d are two rational numbers, such that a/b < c/d ,then 1/2{a/b + c/d} is a rational number lying between a/b and c/d.

Type 1. To find one rational number between two given distinct rational numbers.

Example 1: Find a rational number between 1/3 and 1/4.

Required rational number = 1/2 [1/3 + 1/4] = 1/2[4+3]
                                                                                 12
                                       
                                          = 1/2 x 7/12 = 7/24 .

Type 2. To find many rational numbers between two given distinct rational numbers.

Method :1

Example 2: Find four rational numbers between 1/6 and 1/3.

Solution: Let q1 , q2, q3 and q4 be the four required rational numbers, Then,

q1 = 1/2[1/6 +1/3] = 1/4;

q2 = 1/2[1/4 + 1/3] =7/24;

q3 = 1/2[7/24 +1/3] = 5/16;

q4 = 1/2[5/16 + 1/3] = 31/96.

Hence, four rational numbers between 1/6 and 1/3 are 1/4, 7/24, 5/16 and 31/96.

Method :2

We can find the required number of rational numbers by finding the equivalent rational numbers,having a sufficiently large common denominator depending on how many rational numbers, we have to find.

Example 3: Find (i) 7 rational numbers, (ii) 70 rational numbers, (iii) 100 rational numbers between 1/3 and 3/5.

Solution:

(i) Taking the equivalent rationals, 10/30 for 1/3 and 18/30 for 3/5 , we find that seven integers between 10 and 18 are 11,12,13,14,15,16 and 17.
therefore,  10/30 < 11/30 < 12/30  < ...........< 16/30 < 17/30 < 18/30.

                            i.e.,1/3 < 11/30 < 12/30  < ...........< 16/30 < 17/30 < 3/5.

Hence, the seven required rational numbers between 1/3 and 3/5 are 11/30, 12/30,...... , 17/30.

(ii) Taking the equivalent rational numbers 100/300 for 1/3 and 180/300 for 3/5, we find that the  integers lying between  100 and 180  are 101,102, 103, .............., 179.

Therefore, the seventy rational numbers between 1/3 and 3/5 , may taken as  101/300, 102/300, ...........,170/300.

 (iii) Taking the equivalent rational numbers 1000/3000 for 1/3 and 1800/3000 for 3/5, we may have the 100 integers between 1000 and 1800 as 1001,1002,................,1100.

Hence, the 100 rational numbers between 1/3 and 3/5 are 1001/3000,1002/3000,............1099/3000, 1100/3000.

NOTE: By taking a larger and larger common denominator, we can find more and more rational numbers between 1/3 and 3/5.Suppose we take 1/3 = 10,000/30,000 and 3/5 = 18,000/30,000  then we can find more than 1000 rational numbers between 1/3 and 3/5.


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See you in the next class!!