DENSENESS PROPERTY OF RATIONAL NUMBER –
Theorem – Between two rational numbers x & y, there is always a rational number
(a + b)/2 such that a < (a + b)/2 < b
Proof .- a < b => (a + a) < (a + b) & (a + b) < (b + b)
=> 2a < (a + b) & (a + b) < 2b
=> a < (a + b)/2 < b
Since the sum of two rational numbers is a rational number and the product of two rational numbers is a rational number.
So, (a + b)/2 is a rational number such that a < (a + b)/2 < b
By, the repeated use of the above property, we can say that there are infinitely many rational numbers between two rational numbers.
Example.1) Find out three rational numbers between 5 & 7
Ans.) 5 < 7 => 5 < (5 + 7)/2 < 7
=> 5 < 6 < 7
=> 5 < (5 + 6)/2 < 6 < (6 + 7)/2 < 7
=> 5 < 11/2 < 6 < 13/2 < 7
11 6 13
So, three rational number between 5 & 7 are ------, ------, ------
2 1 2
(Ans.)
Example.2) Find out three rational numbers between 1/2 & 2/3
1 1 1 2 2
Ans.) 1/2 < 2/3 => ------ < ------ X ( ----- + ----- ) < ------
2 2 2 3 3
1 7 2
=> ------- < ------- < -------
2 12 3
1 1 1 7 7 1 7 2 2
=> ---- < ---- X (---- + ----) < ---- < ---- X (---- + ----) < ----
2 2 2 12 12 2 12 3 3
1 13 7 15 2
=> ------ < ------- < ------- < ------- < -------
2 24 12 24 3
11 6 13
So, three rational number between 5 & 7 are ------, ------, -------
2 1 2
(Ans.)
-3 1
Example.3) Find ten rational numbers between ------ and ------ by
4 6
the short cut method.
Ans.) Short cut method -
LCM of 4 & 6 is 12
-3 -3 3 -9
So, ------- = ------ X ------ = -------
4 4 3 12
1 1 2 2
And, ------ = ------ X ------ = ------
6 6 2 12
-9 -8 -7 -6 -5 -4
Now, ------ < ------ < ------ < ------ < ------ < ------ <
12 12 12 12 12 12
-3 -2 -1 1 2 3
< ----- < ------ < ------ < 0 < ------ < ------ < ------
12 12 12 12 12 12
-9 2
Thus, ten rational numbers between ------ and ------
12 12
-8 -7 -6 -5 -4 -3 -2 -1 1 2
----, ----, ----, ----, ----, ----, ----, ----, 0 , ----, -----
12 12 12 12 12 12 12 12 12 12
(Ans.)
Example.4) Find 9 rational numbers between 0 and 0.2
Ans.) Short cut method –
Let a = 0, and b = 0.2, take n = 9
(b – a) (0.2 – 0) 0.2
Use d = --------- = ---------- = ------- = 0.02
(n + 1) (9 + 1) 10
Required rational number is -
=> (a + d), (a + 2d), (a + 3d), (a + 4d),………….., (a + 9d)
=> 0, (0 + 0.02), (0 + 0.04), (0 + 0.06), (0 + 0.08), (0 + 0.10),………….., (0 + 0.20)
=> 0, 0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18
So, 9 rational numbers between 0 and 0.2 are –
0.02, 0.04, 0.06, 0.08, 0.1, 0.12, 0.14, 0.16, 0.18 (Ans.)