Rational Numbers for Class 7 Maths CBSE | ICSE Board.

 

There are many situations that involve fractional numbers. You can represent a distance of 750m above sea level as 3/4 km. Can we represent 750m below sea level in km? Can we denote the distance of

3/4 km below sea level by -3/4  ? We can see -3/4  is neither an integer, nor a fractional number. We need to extend our number system to rational numbers.

WHAT ARE RATIONAL NUMBERS?

The word ‘rational’ arises from the term ‘ratio’. You know that a ratio like 3:2 can also be written as 3/2 . Here, 3 and 2 are natural numbers. Similarly, the ratio of two integers p and q (q ≠ 0), i.e., p:q can be written in the form p/q . This is the form in which rational numbers are expressed.

A rational number is defined as a number that can be expressed in the form p/q , where p and q are integers and q ≠ 0.

Thus, 4/5 is a rational number. Here, p = 4 and q = 5.

Is -3/4  also a rational number? Yes, because p = – 3 and q = 4 are integers. You have seen many fractions like 3/8, 4/8 ,1, 2/3  etc.

All fractions are rational numbers.

Can you say why? How about the decimal numbers like 0.5, 2.3, etc.? Each of such numbers can be written as an ordinary fraction and, hence, are rational numbers. For example, 0.5 = 5/10.

Any integer can be thought of as a rational number.

For example, the integer – 5 is a rational number, because you can write it as -5/1 .

The integer 0 can also be written as 0 = 0/2,  0/7  or etc. Hence, it is also a rational number.

Thus, rational numbers include integers and fractions.

Equivalent rational numbers

A rational number can be written with different numerators and denominators. For example, consider the rational number – 2/3 .

– 2/3 = –2 x 2 /3 x 2 . We see that – 2/3 is the same as – 4/6 . Also,

–2/3 =-2 x 5/3 x 5 =-10/15. So, – 2/3 is also the same as -10/15.

A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1.

If a rational number is not in the standard form, then it can be reduced to the standard form

EXAMPLE :

 Reduce to standard form: (i) 36 /−24 (ii) -3/-15

SOLUTION

(i) The HCF of 36 and 24 is 12.

Thus, its standard form would be obtained by dividing by –12.

  =   Ans.

 (ii) The HCF of 3 and 15 is 3.

Thus, =  Ans.

 

 RATIONAL NUMBERS BETWEEN TWO RATIONAL NUMBERS

Example :

Reshma took two rational numbers -  and. - 

She converted them to rational numbers with same denominators.

That means denominator of LHS must be equal to RHS

Let’s see,

 =  and the next rational number is  =  =

Now the rational numbers between -3/5 and -1/3 is –

  ,  ,

Example :

Write any 3 rational numbers between –2 and 0.

Solution:    –2 can be written as -  = -  and 0 as - = -  .
Thus we have -  ,   ,  - ,   ,  - ,  …..  between –2 and 0. You can take any three of these

You can do it by finding mean and using

WHAT  WE DISCUSSED TODAY?

1.    A number that can be expressed in the form p/q , where p and q are integers and q ≠ 0, is called a rational number. The numbers 2/3, 3/7, 8  etc. are rational numbers.

2.    All integers and fractions are rational numbers.

3.    If the numerator and denominator of a rational number are multiplied or divided by a non-zero integer, we get a rational number which is said to be equivalent to the given rational number.

For example So, we say -6 /14  is the equivalent form of -3/7 = -3x2/7x2= -6/14 .

Also note that -6/2/14/2 =  -3/7.

4.    Rational numbers are classified as Positive and Negative rational numbers. When the numerator and denominator, both, are positive integers, it is a positive rational number. When either the numerator or the denominator is a negative integer, it is a negative rational number. For example, 3/8 is a positive rational number whereas -8/9  is a negative rational number.

5.    The number 0 is neither a positive nor a negative rational number.

6.    A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1.

The numbers -1/3 , 2/7  etc. are in standard form.

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

8.    Two rational numbers with the same denominator can be added by adding their numerators, keeping the denominator same. Two rational numbers with different denominators are added by first taking the LCM of the two denominators and then converting both the rational numbers to their equivalent forms having the LCM as the denominator. For example, -2/3+3/8  

9.    While subtracting two rational numbers, we add the additive inverse of the rational number to be subtracted to the other rational number.

10.     To multiply two rational numbers, we multiply their numerators and denominators separately and write the product as

 .

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