Rational Numbers

Rational Number: A number that can be expressed in the form of p/q where p and q are integers and q≠0, is called a rational number.
All integers and fractions are rational numbers.

If the numerator and denominator of a rational number are multiplied or divided by a non-zero integer then we get a rational number which is equivalent to the given rational number. Following is an example;

`2/3=(2xx3)/(3xx3)=6/9`

When both numerator and denominator of a rational number are positive integers then the given rational number is a positive rational number. When either the numerator or the denominator is a negative integer then the given rational number is a negative rational number.

The number zero is neither a positive nor a negative rational number. But zero is a rational number because it can be written in the form of p/q where q≠0.

Standard form of Rational Number: When the denominator is a positive integer and both the numerator and denominator are co-prime then the rational number is said to be in the standard form.

There are unlimited number of rational numbers between two rational numbers.

Exercise 9.1

Question 1: List five rational numbers between following rational numbers:

(a) – 1 and 0

Answer: Since we need to find five rational numbers between the given rational numbers, let us write the given numbers with denominator 6.

`-1=(-6)/(6)` and `0=0/6`

Rational numbers between these numbers can be as follows:

`-5/6`, `-4/6`, `-3/6`, `-2/6`, `-1/6`

(b) – 2 and – 1

Answer: Let us write the given numbers with denominator 6:

`-2=(-12)/(6)` and `-1=(-6)/(6)`

Rational numbers between these numbers can be as follows:

`-(11)/(6)`, `(-10)/(6)`, `-9/6`, `-8/6`, `-7/6`

Answer: LCM of denominators (5 and 3) is 15. Let us write the given numbers with 15 as denominator.

`-4/5=-(12)/(15)` and `-2/3=-(10)/(15)`

So, the rational number between two given number is as follows:

`-(11)/(15)`

To find more rational numbers in between, let us change the denominator to 30 (multiply by 2). Then the numbers can be written as follows:

`-(24)/(30)`, `-(22)/(30)`, `-(20)/(30)`

This will help us in writing following rational numbers in between:

`-(23)/(30)` and `-(21)/(30)`

To find more rational numbers in between, let us change the denominator to 60 (multiply by 2). Then the numbers can be written as follows:

`-(48)/(60)`, `-(46)/(60)`, `-(44)/(60)`, `-(42)/(60)`, `-(40)/(60)`

This will help is in writing following rational numbers in between:

`-(47)/(60)`, `-(46)/(60)`, `-(45)/(60)`, `-(44)/(60)`, `-(42)/(60)`

(e) `1/2` and `2/3`

Answer: LCM of denominators (2 and 3) is 6. Converting the given numbers with 6 as denominator gives us following numbers:

`1/2=3/6` and `2/3=4/6`

Above pair has no number between integers in numerators, i.e. 3 and 4.

So, let us convert these numbers to get denominator 12:

`1/2=(6)/(12)` and `2/3=(8)/(12)`

Above pair gives scope for one rational number in between. So, let us convert these numbers to get denominator 24:

`1/2=(12)/(24)` and `2/3=(16)/(24)`

Above pair gives scope for three rational numbers in between. So, let us convert these numbers to get denominator 48:

`1/2=(24)/(48)` and `2/3=(32)/(48)`

Now, required rational numbers as are follows:

`(25)/(48)`, `(26)/(48)`, `(27)/(48)` and `(29)/(48)`

Question 2: Write four more rational numbers in each of the following patterns:

(a) `-3/5`, `-(6)/(10)`, `-(9)/(15)`, `-(12)/(20)`

Answer: The series is increasing by multiplying the numerator and denominator of the first number subsequently by 2, 3, 4, etc. So, next four rational numbers in the series are:

`-(15)/(25)`, `-(18)/(30)`, `-(21)/(35)`, `-(24)/(40)`

(b) `-1/4`, `-2/8`, `-(3)/(12)`

Answer: Next four rational numbers in the series are:

`-(4)/(16)`, `-(5)/(20)`, `-(6)/(24)`, `-(7)/(28)`

Answer: `(8)/(-12)`, `(10)/(-15)`, `(12)/(-18)`, `(14)/(-21)`

Question 3: Give four rational numbers equivalent to following rational numbers:

(a) `-2/7`

Answer: `-(4)/(14)`, `-(6)/(21)`, `-(8)/(28)`, `-(10)/(35)`

(b) `(5)/(-3)`

Answer: `(10)/(-6)`, `(15)/(-9)`, `(20)/(-12)`, `(25)/(-15)`

Answer: `(8)/(18)`, `(12)/(27)`, `(16)/(36)`, `(20)/(45)`

Question 4: Draw the number line and represent the following rational numbers on it: `3/4`, `-5/8`, `-7/4` and `7/8`

Answer: