Number System: 9 Math
NCERT Exercise 1.1 -Rational Number: 9th math
Numbers which can be written in the form of `p/q` where p and q are integers and q≠0 are called RATIONAL NUMBERS.
Example
(a) 2 can be written as rational number i.e. as 2/1, i.e. in the form of `p/q`
where p = 2 and q= 1 are integers and q≠0. Thus, 2 is a rational number.
(b) – 2 can also be written as rational number i.e. as `(-2)/1`, i.e. in the form of p/q where p= –2 and q = 1 are integers and q≠0. Thus, –2 is a rational number.
(c) 5/2 is also a rational number. Because it is in the form of `p/q` where, p= 5 and q =2 are integers and q≠0.
Hence,
All Natural Numbers, i.e. 1, 2, 3, 4, 5, . . . . can be written in the form of `p/q`, where, q≠0.
Thus, all Natural Numbers are Rational Number.
All Whole Numbers, i.e. 0, 1, 2, 3, 4, . . . . can be written in the form of `p/q`, where, q≠0.
Thus, every Whole Numbers is a Rational Number.
All Integers can be written in the form of `p/q` where, q≠0.
Thus, all Integers are Rational Number.
Equivalent Rational Numbers
A Rational Number do not have a unique representation in the form of `p/q`, where p and q are integers and q≠0.
Example:
`1/2=2/4=3/6=25/50`
and so on.
Since, these given numbers are equal and are in the form of `p/q` where p and q are integers and q≠0. Thus, these are Equivalent Rational Numbers or Equivalent Fractions.
But when it is said that, `p/q` is a rational number, or when `p/q`
is represented on the number line, it is assumed that q≠0 and p and q have no common factors other than 1. This, means p and q are co-prime.
So on the Number Line, among the infinitely many fractions equivalent to `1/2`, but we will choose `1/2` to represent all of them.
Solution of NCERT Exercise 1.1 (class ninth mathematics)
Question (1) Is zero a rational number? Can you write it in the form p/q, where p and q are integers and q≠0.
Answer:
Yes zero is a rational number.
Zero (0) can be written in the form of `p/q`
i.e. `o=0/1`
Here, p=0 and q=1 are integers and q=1≠0.
Question: (2) Find six rational numbers between 3 and 4.
Solution:
1st method
We know, that a rational number lies between middle of two number can be found after dividing the sum of given numbers by 2.
This means a rational number between given two numbers can be found by calculating the average of given numbers.
Here, given numbers = 3 and 4
(a) 1st rational number between 3 and 4
Average of 3 and 4
`=(3+4)/2=7/2`
(b) 2nd Rational Number between 3 and 4
The Rational Number between `7/2` and 4 will be another Rational number between 3 and 4.
Thus, average of `7/2` and 4
`=(7/2+4)//2=((7+8)/2)//2`
`=(15//2)/2=15/(2xx2)=15/4`
Thus, `15/4` is the another Rational number between given numbers 3 and 4
(c) 3rd Rational Number between 3 and 4
The Rational Number between `15/4` and 4 will be one of the Rational Numbers between given numbers 3 and 4.
Thus, rational number between `15/4` and 4
`=(15/4+4)//2 =((15+16)4)//2`
`=(31/4)//2=31/(4xx2)=31/8`
Thus, 31/8 is another Rational Number between given numbers 3 and 4
(d) 4th Rational Number between 3 and 4
The Rational Number between `31/8` and 4 will be one of the Rational Numbers between given numbers 3 and 4. Because, `31/8` is one of the Rational numbers between 3 and 4.
Thus, rational number between `31/8` and 4
`=(31/8+4)//2 =((31+32)/8)//2`
`=(63/8)//2=63/(8xx2)=63/16`
Thus, `63/16` is another Rational Number between given numbers 3 and 4
(e) 5th Rational Number between 3 and 4
The Rational Number between `63/16` and 4 will be one of the Rational Numbers between given numbers 3 and 4. Because, `63/16` is one of the Rational numbers between 3 and 4.
Thus, rational number between `63/16` and 4
`=(63/16+4)//2 =((63+64)/16)//2`
`=(127/16)//2=127/(16xx2)=127/32`
Thus, `127/32` is another Rational Number between given numbers 3 and 4
(f) 6th Rational Number between 3 and 4
The number between `127/32` and 4 will be one of the Rational Numbers between given numbers 3 and 4. Because, `127/32` is one of the Rational numbers between 3 and 4.
Thus, number between `127/32` and 4
`=(127/32+4)//2=((127+128)/32)//2`
`=(255/32)//2=255/(32xx2)=255/64`
Thus, 255/64 is another Rational Number between given numbers 3 and 4
Thus, `7/2, 15/4, 31/8, 63/16, 127/32` and `255/64` are six Rational Numbers between 3 and 4 Answer
Alternate Method to find six Rational Numbers between given numbers 3 and 4
Given numbers = 3 and 4
Since, we have to find 6 Rational numbers between 3 and 4
Thus, we can write 3 and 4 as Rational Numbers with denominator = 6+1 = 7
Thus, `3=(3xx7)/7 =21/71
And, `4=(4xx7)/7=28/7`
Now, we have to find six rational numbers between `21/7` and `28/7`
Thus, six rational numbers between `21/7` and `28/7` equal to
`22/7, 23/7, 24/7, 25/7, 26/7` and `27/7` Answer
Question (3) Find five rational numbers between `3/5` and `4/5`
Solution:
1st Method
Finding Rational numbers between given two numbers using finding Average between given rational numbers .
Given, `3/5` and `4/5`
(a) Calculation of First rational number between `3/5` and `4/5`
Average of `3/5` and `4/5`
`=(3/5+4/5)//2`
`=((3+4)/5)//2`
`=(7//5)/2`
`=7/(5xx2)`
`=7/10`
(b) Calculation of Second Rational Number between `3/5` and `4/5`
Since, `7/10` is one of the rational numbers between `3/5` and `4/5`.
Thus, rational number between `7/10` and `4/5` will be one of the rational numbers between `3/5` and `4/5`.
Thus, Average of `7/10` and `4/5`
`=(7/10+4/5)//2`
`=((7+8)/10)//2`
`=(15/10)//2`
`=15/(10xx2)`
`=15/20` ----------(i)
`= (5xx3)/(5xx4)`
`=5/4`
(c) Calculation of Third Rational Number between `3/5` and `4/5`
Since, `15/20` is one of the rational numbers between `3/5` and `4/5`.
[From equation (i)]
Thus, rational number between `15/20` and `4/5` will be one of the rational numbers between `3/5` and `4/5`.
Thus, Average of `15/20` and `4/5`
`=(15/20+4/5)//2`
`=((15+16)/20)//2`
`=(31/20)//2`
`=31/(20xx2)`
`=31/40`
(d) Calculation of Forth Rational Number between `3/5` and `4/5`
Since, `31/40` is one of the rational numbers between `3/5` and `4/5`.
Thus, rational number between `31/40` and `4/5` will be one of the rational numbers between `3/5` and `4/5`.
Thus, Average of `31/40` and `4/5`
`=(31/40+4/5)//2`
`=((31+32)/40)//2`
`=(63/40)//2`
`=63/(40xx2)`
`=63/80`
(e) Calculation of Fifth Rational Number between `3/5` and `4/5`
Since, `63/80` is one of the rational numbers between `3/5` and `4/5`.
Thus, rational number between `63/80` and `4/5` will be one of the rational numbers between `3/5` and `4/5`.
Thus, Average of `63/80` and `4/5`
`=(63/80+4/5)//2`
`=((63+64)/80)//2`
`=(127/80)//2`
`=127/(80xx2)`
`=127/160`
Thus, `7/10, 15/20, 31/40, 63/80` and `127/160` are five rational numbers between `3/5` and `4/5` Answer
Alternate Method to find required Rational Numbers between given rational numbers
Given, rational numbers `3/5` and `4/5`
Five rational numbers between given numbers = ?
Since, we have to find 5 rational numbers between `3/5` and `4/5`.
Thus, we will write `3/5` and `4/5` as rational numbers with denominator 5+1 = 6
Now, `3/5 = (3xx6)/(5xx6)=18/30`
And, `4/5 = (4xx6)/(5xx6) = 24/30`
Now, five rational numbers between `18/30` and `24/30` equal to
`19/30, 20/30, 21/30, 22/30` and `23/30`
Since, `20/30 = 2/3`
And, `21/30 = (7xx3)/(10xx3)=7/10`
And, `22/30 = (11xx2)/(15xx2) = 11/15`
Thus, five rational numbers between `18/30` and `24/30` equal to
`19/30, 2/3, 7/10, 11/15` and `23/30` Answer
Question (4) State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
Answer: True
Reason: Natural Numbers along with zero are called Whole Numbers. Thus every natural numbers is a whole number.
(ii) Every integer is a whole number
Answer: False
Reason: All natural numbers, zero and negative of counting numbers collectively form Integers. While whole numbers are natural numbers along with zero.
Since whole numbers do not contain negative numbers while integers can be negative too. Thus every integer is not a whole number.
(iii) Every rational number is a whole number
Answer: False
Reason: Whole numbers do not contain fractions while rational numbers contain fraction too. Thus, every rational number is not a whole number.
Example: `1/2` is a rational number while it is not a whole number.
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