2. Rational numbers and operations on rational

 

2. Rational numbers and operations on rational

Learn and Remember:

1. If p is any integer and q is any non-zero integer, then  ^p ⁄ _q  is a rational number. The numerator of this rational number is p, and the denominator is q.

2. If the numerator and the denominator of a rational number carry the same signs, the number is a positive rational number. If the numerator and the denominator of a rational number carry different signs, the number is a negative rational number.

 e.g. ⁹ ⁄ ₅ , ^-4 ⁄ _-7  =: - are positive rational numbers,

  ^{- 3} ⁄ _{9 ,}  ⁶ ⁄ _{-11  =}  are negative rational numbers.

 3. The equivalent rational numbers can be obtained by multiplying the numerator and the denominator of the given rational number by the same non-zero integer.

e.g. ¹ ⁄ ₂= ³ ⁄ ₆= ⁴ ⁄ ₈...........

4. If we go on reducing the given rational number, we reach a stage when there is no common factor except 1 in its numerator and denominator. In such a case, we say that the rational number is reduced to its lowest terms.

     To reduce a rational number to its lowest terms, its numerator and denominator should be divided by their HCF (GCD).

5. Order relation between two Rational Numbers :

To find the order relation between two rational numbers, first make their denominators positive and then convert them into equivalent rational numbers having the same denominator.

 Let us find the order relation between two rational numbers and  ^a ⁄ _b and ^c ⁄ _d  Make the

denominators b and d positive.

^a ⁄ _b= ^ad ⁄ _{b d} and ^c ⁄ _d = ^bc ⁄ _{b d}  We have obtained two equivalent rational number ^ad ⁄ _{b d} and ^bc ⁄ _{b d} 

Since, their denominators are equal, their order relation depends on the numerators ad and bc.

(i) If ad = bc, then ^a ⁄ _{b = ^c ⁄ d}

(ii) If ad > bc, then  ^a ⁄ _{b > ^c ⁄ d}

(iii) If ad <bc, then ^a ⁄ _{b < ^c ⁄ d}

Follow this method to find the order relation between any two rational numbers.

(iv) Rational numbers are infinite. There is no smallest or the greatest rational number.

(v) If the denominators of positive rational numbers are the same, the rational number with greater numerator is greater.

⁷ ⁄ ₁₂ >⁵ ⁄ ₁₂ >¹ ⁄ _12,

(vi) If the denominators of negative rational numbers are the same, the rational number with smaller numerator is greater.

e.g. ^-3 ⁄ ₄ > ^-5 ⁄ ₄ >^-7 ⁄ ₄

6. Addition of Rational Numbers :

First write the rational numbers with positive denominators and then add them by the same procedure as for adding fractions.

7. Subtraction of Rational Numbers :

If the sum of two rational numbers is zero, then they are called opposites of each other. Subtracting one rational number from the other means adding its opposite rational number to the other. 

e.g. ⁴⁄₅-²⁄₃=⁴⁄₅+ (-²⁄₃₎

 8. Multiplication of Rational Numbers :

(i) Product of two positive or two negative rational numbers is positive.

(ii) Product of a positive rational number and a negative rational number is negative.

(iii) Given two or more rational numbers, product of the numerators divided by product of the denominators is equal to the product of the given rational numbers. Simplification can be done at an earlier stage to reduce the answer to the lowest terms. This avoids big calculations.

^{e.g.  3}⁄₅  x  ¹⁄₆  x  ¹⁵⁄_{17  = ³⁄34}

9. The product of the sum of two rational numbers with a third rational number is equal to the sum of the products of the two rational numbers with the third.

e.g. ¹⁄₂ ( ¹⁄₃ + ⁵⁄₆ ) = ¹⁄₂   x  ¹⁄₃+  ¹⁄₂  x  ⁵ ⁄₆

10. Division of Rational Numbers :

(i) If the product of two rational numbers is 1, each is called the multiplicative inverse orreciprocal of the other.

(ii) To divide a rational number by any other non-zero rational number, we should multiply the first rational number by the reciprocal of the second rational number.

11. (i) The sum, difference, product or quotient of two rational numbers is a rational number (ii) There are infinite rational numbers between any two rational numbers.