CBSE Class 8 Maths Advanced Rational Numbers Notes Set 01

Rational Numbers

Rational Numbers

Numbers fall into two big families: real numbers and imaginary numbers. Real numbers are further split into rational and irrational numbers.

Real Numbers

A real number is one that can stand for an actual physical quantity, such as a length or a mass. Every real number can be placed on the number line, which is a straight line with a point fixed as zero (the origin).

Natural Numbers

The counting numbers form the set of natural numbers, N = { 1, 2, 3, 4, ... }.

Whole Numbers

Adding zero to the natural numbers gives the whole numbers, W = { 0, 1, 2, 3, 4, ... }.

Prime Numbers

A natural number having exactly two different factors, 1 and the number itself, is prime. In other words it is divisible by nothing except 1 and itself. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23, ...

Composite Numbers

A natural number greater than 1 that is not prime is composite, for example 4, 6, 9, 10. Remark: the number 1 is neither prime nor composite.

Integers

The natural numbers, zero and the negatives of the natural numbers together form the integers, Z (or I) = { ..., −3, −2, −1, 0, 1, 2, 3, ... }.

Identifying a Prime Number

Step 1: estimate the square root of the given number. Step 2: divide the number by every prime smaller than that estimate. If none of these primes divides it exactly, the number is prime; otherwise it is not.

Illustration 1.1. Is 571 a prime number ?
Answer:
The square root of 571 is about 24.
The primes below 24 are 2, 3, 5, 7, 11, 13, 17, 19 and 23. None of them divides 571 exactly.
So 571 is a prime number.

Rational Numbers (Definition)

A number that can be written as \( \frac{p}{q} \) with p, q integers and q ≠ 0 is a rational number. Each of \( \frac{2}{3} \), \( \frac{-5}{7} \), \( \frac{-11}{-5} \) and \( \frac{7}{-9} \) is rational.

Positive rational number. \( \frac{p}{q} \) is positive when p and q have the same sign (both positive or both negative), for example \( \frac{2}{3} \), \( \frac{5}{9} \), \( \frac{-7}{-12} \), \( \frac{-3}{-11} \).

Negative rational number. \( \frac{p}{q} \) is negative when p and q have opposite signs, for example \( \frac{-3}{7} \), \( \frac{5}{-9} \), \( \frac{-15}{26} \).

Decimal Representation of Rational Numbers

Terminating decimals. If the decimal expansion of a rational number \( \frac{p}{q} \) ends after some digits, then the prime factorisation of q has the form \( 2^m \times 5^n \) (m, n non-negative integers). For example \( \frac{1}{2} \) = 0.5 and \( \frac{3}{20} \) = 0.15.

Non-terminating repeating (recurring) decimals. If the prime factorisation of q is not of the form \( 2^m \times 5^n \), the decimal never ends but a block of digits repeats forever. For example \( \frac{2}{3} \) = 0.6666... = \( 0.\overline{6} \) and \( \frac{5}{11} \) = 0.4545... = \( 0.\overline{45} \).

Lowest Form of a Rational Number

\( \frac{p}{q} \) is in lowest (simplest) form when p and q have no common factor other than 1. To reduce a rational number: (1) take \( \frac{p}{q} \); (2) find the HCF m of p and q; (3) if m = 1 it is already in lowest form; (4) if m ≠ 1 divide both p and q by m, giving \( \frac{p ÷ m}{q ÷ m} \).

Illustration 1.2. Express each of the following rational numbers in the lowest form. (i) \( \frac{12}{16} \) (ii) \( \frac{-60}{72} \)
Answer:
(i) 12 = 2 × 2 × 3 and 16 = 2 × 2 × 2 × 2, so HCF = 2 × 2 = 4. Dividing top and bottom by 4: \( \frac{12}{16} \) = \( \frac{12 ÷ 4}{16 ÷ 4} \) = \( \frac{3}{4} \). So \( \frac{3}{4} \) is the lowest form.
(ii) 60 = 2 × 2 × 3 × 5 and 72 = 2 × 2 × 2 × 3 × 3, so HCF = 2 × 2 × 3 = 12. Dividing by 12: \( \frac{-60}{72} \) = \( \frac{-5}{6} \).

Note.

1. Two rational numbers are equal if they have the same lowest (standard) form.
2. If \( \frac{x}{y} \) is rational and m is any non-zero integer, then \( \frac{x}{y} \) = \( \frac{x \times m}{y \times m} \). Example: \( \frac{3}{8} \) = \( \frac{3 \times 4}{8 \times 4} \) = \( \frac{12}{32} \).
3. If m is a common divisor of x and y, then \( \frac{x}{y} \) = \( \frac{x ÷ m}{y ÷ m} \). Example: \( \frac{-27}{45} \) = \( \frac{(-27) ÷ 3}{45 ÷ 3} \) = \( \frac{-9}{15} \) = \( \frac{(-9) ÷ 3}{15 ÷ 3} \) = \( \frac{-3}{5} \).

Representation of Rational Numbers on the Number Line

Draw a line and mark a point 0 on it. Using one fixed length as a unit, step off equal distances to the right and to the left of 0. The points to the right are A, B, C, D, E for the integers 1, 2, 3, 4, 5, and the points to the left are A', B', C', D', E' for −1, −2, −3, −4, −5. Positive integers lie to the right of 0 and negative integers to the left. Rational numbers are shown in a similar way.

Illustration 1.3. Represent (a) \( \frac{2}{5} \) (b) \( \frac{-7}{3} \) on the number line.
Answer:
(a) Draw a number line, mark O for 0 and A for 2. Divide OA into 5 equal parts (the denominator of \( \frac{2}{5} \)) using points P, Q, R, S. The first point P stands for \( \frac{2}{5} \).

(b) Mark O for 0 and A' for −7, which is 7 units to the left of O. Divide OA' into 3 equal parts at P' and Q'. The point P' next to O stands for \( \frac{-7}{3} \).

Absolute Value

The absolute value of a rational number is its numerical value without the sign. For example \( \left| -\frac{3}{5} \right| = \frac{3}{5} \) and \( \left| \frac{7}{9} \right| = \frac{7}{9} \).

Illustration 1.4. Verify that |x + y| ≤ |x| + |y| by taking x = \( \frac{3}{5} \), y = \( \frac{-4}{15} \)
Answer:
x + y = \( \frac{3}{5} \) + \( \frac{-4}{15} \) = \( \frac{9 - 4}{15} \) = \( \frac{5}{15} \), so |x + y| = \( \frac{5}{15} \).
|x| + |y| = \( \frac{3}{5} \) + \( \frac{4}{15} \) = \( \frac{9 + 4}{15} \) = \( \frac{13}{15} \).
Since \( \frac{5}{15} \) < \( \frac{13}{15} \), the statement |x + y| ≤ |x| + |y| holds.

Illustration 1.5. Verify that |x × y| = |x| × |y| by taking x = \( \frac{-5}{3} \), y = \( \frac{7}{9} \)
Answer:
|x × y| = \( \left| \frac{-5}{3} \times \frac{7}{9} \right| \) = \( \frac{35}{27} \).
|x| × |y| = \( \frac{5}{3} \) × \( \frac{7}{9} \) = \( \frac{35}{27} \).
Hence |x × y| = |x| × |y|.

Comparing Two Rational Numbers

Steps: (1) take the two numbers; (2) rewrite each so that its denominator is positive; (3) find the LCM of the (positive) denominators; (4) rewrite both numbers with the LCM as common denominator; (5) compare numerators: the one with the larger numerator is the larger number.

Illustration 1.6. Which of the two rational numbers \( \frac{3}{5} \) and \( \frac{-2}{3} \) is greater ?
Answer:
The first is positive and the second is negative, and a positive rational number is always greater than a negative one.
So \( \frac{3}{5} \) > \( \frac{-2}{3} \).

Illustration 1.7. Which of the two rational numbers \( \frac{5}{7} \) and \( \frac{3}{5} \) is greater ?
Answer:
Both denominators are positive. LCM of 7 and 5 is 35.
\( \frac{5}{7} \) = \( \frac{5 \times 5}{7 \times 5} \) = \( \frac{25}{35} \) and \( \frac{3}{5} \) = \( \frac{3 \times 7}{5 \times 7} \) = \( \frac{21}{35} \).
Since 25 > 21, \( \frac{5}{7} \) > \( \frac{3}{5} \).

Illustration 1.8. Arrange the rational numbers \( \frac{-7}{10} \), \( \frac{5}{-8} \), \( \frac{2}{-3} \) in ascending order.
Answer:
Make the denominators positive: \( \frac{5}{-8} \) = \( \frac{-5}{8} \) and \( \frac{2}{-3} \) = \( \frac{-2}{3} \). The numbers are \( \frac{-7}{10} \), \( \frac{-5}{8} \), \( \frac{-2}{3} \).
LCM of 10, 8, 3 = 120: \( \frac{-7}{10} \) = \( \frac{-84}{120} \), \( \frac{-5}{8} \) = \( \frac{-75}{120} \), \( \frac{-2}{3} \) = \( \frac{-80}{120} \).
Since −84 < −80 < −75, the order is \( \frac{-7}{10} \) < \( \frac{-2}{3} \) < \( \frac{-5}{8} \), that is \( \frac{-7}{10} \) < \( \frac{2}{-3} \) < \( \frac{5}{-8} \).

Operations on Rational Numbers

Addition

Before adding, write each rational number with a positive denominator.

Case I: same denominators. \( \frac{a}{b} \) + \( \frac{c}{b} \) = \( \frac{a + c}{b} \).

Illustration 1.9. Add \( \frac{7}{5} \) and \( \frac{9}{5} \)
Answer:
\( \frac{7}{5} \) + \( \frac{9}{5} \) = \( \frac{7 + 9}{5} \) = \( \frac{16}{5} \).

Case II: different denominators. Take the LCM of the denominators, convert each number to that denominator, then add as in Case I.

Illustration 1.10. Add \( \frac{3}{8} \) and \( \frac{5}{6} \)
Answer:
LCM of 8 and 6 is 24. \( \frac{3}{8} \) = \( \frac{3 \times 3}{8 \times 3} \) = \( \frac{9}{24} \) and \( \frac{5}{6} \) = \( \frac{5 \times 4}{6 \times 4} \) = \( \frac{20}{24} \).
Sum = \( \frac{9}{24} \) + \( \frac{20}{24} \) = \( \frac{29}{24} \).

Short-cut method. Divide the LCM by each denominator, multiply the numerator by that quotient, and add the results over the LCM.

Illustration 1.11. Find the sum \( \frac{-7}{5} \) + \( \frac{2}{3} \)
Answer:
LCM of 5 and 3 = 15.
\( \frac{-7}{5} \) + \( \frac{2}{3} \) = \( \frac{3 \times (-7) + 5 \times 2}{15} \) = \( \frac{-21 + 10}{15} \) = \( \frac{-11}{15} \).

Properties of Addition

1. Closure. The sum of two rational numbers is again a rational number. For example \( \frac{1}{3} \) + \( \frac{3}{4} \) = \( \frac{4 + 9}{12} \) = \( \frac{13}{12} \).

2. Commutative law. Rational numbers can be added in any order: \( \frac{a}{b} \) + \( \frac{c}{d} \) = \( \frac{c}{d} \) + \( \frac{a}{b} \). For example \( \frac{1}{2} \) + \( \frac{3}{4} \) = \( \frac{5}{4} \) = \( \frac{3}{4} \) + \( \frac{1}{2} \).

3. Associative law. Three rational numbers can be grouped in any way: (\( \frac{a}{b} \) + \( \frac{c}{d} \)) + \( \frac{e}{f} \) = \( \frac{a}{b} \) + (\( \frac{c}{d} \) + \( \frac{e}{f} \)). Example with \( \frac{-2}{3} \), \( \frac{5}{7} \), \( \frac{1}{6} \):

(\( \frac{-2}{3} \) + \( \frac{5}{7} \)) + \( \frac{1}{6} \) = \( \frac{1}{21} \) + \( \frac{1}{6} \) = \( \frac{2 + 7}{42} \) = \( \frac{9}{42} \) = \( \frac{3}{14} \).
\( \frac{-2}{3} \) + (\( \frac{5}{7} \) + \( \frac{1}{6} \)) = \( \frac{-2}{3} \) + \( \frac{37}{42} \) = \( \frac{-28 + 37}{42} \) = \( \frac{9}{42} \) = \( \frac{3}{14} \). Both groupings agree.

4. Additive identity. Zero is the additive identity: \( \frac{a}{b} \) + 0 = 0 + \( \frac{a}{b} \) = \( \frac{a}{b} \). For example \( \frac{3}{5} \) + 0 = \( \frac{3}{5} \).

5. Additive inverse. For every \( \frac{a}{b} \) there is \( \frac{-a}{b} \) with \( \frac{a}{b} \) + \( \frac{-a}{b} \) = 0. This number \( \frac{-a}{b} \) is the additive inverse (negative) of \( \frac{a}{b} \). For example \( \frac{4}{7} \) and \( \frac{-4}{7} \) are additive inverses of each other.

Subtraction

To subtract, add the additive inverse: \( \frac{a}{b} \) − \( \frac{c}{d} \) = \( \frac{a}{b} \) + (\( \frac{-c}{d} \)).

Illustration 1.12. Find the additive inverse of (i) \( \frac{5}{9} \) (ii) \( \frac{9}{-11} \)
Answer:
(i) The additive inverse of \( \frac{5}{9} \) is \( \frac{-5}{9} \).
(ii) In standard form \( \frac{9}{-11} \) = \( \frac{-9}{11} \), so its additive inverse is \( \frac{9}{11} \).

Illustration 1.13. Subtract (i) \( \frac{-5}{7} \) from \( \frac{-2}{5} \) (ii) \( \frac{9}{16} \) from \( \frac{7}{24} \)
Answer:
(i) \( \frac{-2}{5} \) − \( \frac{-5}{7} \) = \( \frac{-2}{5} \) + \( \frac{5}{7} \) = \( \frac{-14 + 25}{35} \) = \( \frac{11}{35} \).
(ii) \( \frac{7}{24} \) − \( \frac{9}{16} \) = \( \frac{14 - 27}{48} \) = \( \frac{-13}{48} \).

Illustration 1.14. What number should be added to \( \frac{-7}{8} \) to get \( \frac{4}{9} \) ?
Answer:
Let the number be x. Then \( \frac{-7}{8} \) + x = \( \frac{4}{9} \).
x = \( \frac{4}{9} \) + \( \frac{7}{8} \) = \( \frac{32 + 63}{72} \) = \( \frac{95}{72} \).
The required number is \( \frac{95}{72} \).

Multiplication

The product of two rational numbers is found by multiplying numerators and denominators separately: \( \frac{a}{b} \) × \( \frac{c}{d} \) = \( \frac{a \times c}{b \times d} \).

Illustration 1.15. Find the products (i) \( \frac{-15}{4} \) × \( \frac{-3}{8} \) (ii) \( \frac{3}{7} \) × \( \frac{-5}{8} \)
Answer:
(i) \( \frac{(-15) \times (-3)}{4 \times 8} \) = \( \frac{45}{32} \).
(ii) \( \frac{3 \times (-5)}{7 \times 8} \) = \( \frac{-15}{56} \).

Properties of Multiplication

1. Closure. The product of two rational numbers is a rational number: \( \frac{1}{2} \) × \( \frac{5}{7} \) = \( \frac{5}{14} \).

2. Commutative law. \( \frac{a}{b} \) × \( \frac{c}{d} \) = \( \frac{c}{d} \) × \( \frac{a}{b} \). Example: \( \frac{3}{4} \) × \( \frac{5}{7} \) = \( \frac{15}{28} \) = \( \frac{5}{7} \) × \( \frac{3}{4} \).

3. Associative law. (\( \frac{a}{b} \) × \( \frac{c}{d} \)) × \( \frac{e}{f} \) = \( \frac{a}{b} \) × (\( \frac{c}{d} \) × \( \frac{e}{f} \)). Example with \( \frac{-5}{2} \), \( \frac{-7}{4} \), \( \frac{1}{3} \): both groupings give \( \frac{35}{24} \).

4. Multiplicative identity. 1 is the multiplicative identity: \( \frac{a}{b} \) × 1 = 1 × \( \frac{a}{b} \) = \( \frac{a}{b} \). Example: \( \frac{3}{4} \) × 1 = \( \frac{3}{4} \).

5. Multiplicative inverse. Every non-zero rational number \( \frac{a}{b} \) has an inverse \( \frac{b}{a} \) with \( \frac{a}{b} \) × \( \frac{b}{a} \) = 1. It is also called the reciprocal. Zero has no reciprocal; the reciprocal of 1 is 1 and the reciprocal of −1 is −1. Example: the reciprocal of \( \frac{5}{7} \) is \( \frac{7}{5} \).

6. Distributive law (multiplication over addition). \( \frac{a}{b} \) × (\( \frac{c}{d} \) + \( \frac{e}{f} \)) = (\( \frac{a}{b} \) × \( \frac{c}{d} \)) + (\( \frac{a}{b} \) × \( \frac{e}{f} \)). Example with \( \frac{-3}{4} \), \( \frac{2}{3} \), \( \frac{-5}{6} \): left side = \( \frac{-3}{4} \) × \( \frac{-1}{6} \) = \( \frac{1}{8} \); right side = \( \frac{-1}{2} \) + \( \frac{5}{8} \) = \( \frac{1}{8} \).

7. Multiplication by 0. Any rational number times 0 is 0: \( \frac{5}{18} \) × 0 = 0 = 0 × \( \frac{5}{18} \).

Illustration 1.16. Find the reciprocal of (i) −8 (ii) \( \frac{5}{16} \)
Answer:
(i) The reciprocal of −8 is \( \frac{1}{-8} \) = \( \frac{-1}{8} \).
(ii) The reciprocal of \( \frac{5}{16} \) is \( \frac{16}{5} \).

Division

In \( \frac{a}{b} \) ÷ \( \frac{c}{d} \), the first number is the dividend, the second the divisor and the result the quotient. Dividing means multiplying by the reciprocal of the divisor: \( \frac{a}{b} \) ÷ \( \frac{c}{d} \) = \( \frac{a}{b} \) × \( \frac{d}{c} \).

Properties of division. (1) Closure: if \( \frac{c}{d} \) ≠ 0, then \( \frac{a}{b} \) ÷ \( \frac{c}{d} \) is a rational number. (2) \( \frac{a}{b} \) ÷ 1 = \( \frac{a}{b} \). (3) For a non-zero \( \frac{a}{b} \), \( \frac{a}{b} \) ÷ \( \frac{a}{b} \) = 1.

Illustration 1.17. Divide \( \frac{4}{7} \) by \( \frac{-3}{8} \)
Answer:
\( \frac{4}{7} \) ÷ \( \frac{-3}{8} \) = \( \frac{4}{7} \) × \( \frac{-8}{3} \) = \( \frac{-32}{21} \).

Insertion of Rational Numbers between Two Rational Numbers

If a < b, then \( \frac{1}{2}(a+b) \) is a rational number lying between a and b. Repeating the idea gives as many numbers as wanted; in fact there are infinitely many rational numbers between any two rational numbers. Another way is to give both numbers the same denominator and pick the numerators between them (using larger multiples of the denominator if needed).

Illustration 1.18. Find 3 rational numbers between \( \frac{1}{3} \) and \( \frac{1}{2} \)
Answer:
Between \( \frac{1}{3} \) and \( \frac{1}{2} \): \( \frac{1}{2}\left(\frac{1}{3}+\frac{1}{2}\right) \) = \( \frac{5}{12} \).
Between \( \frac{1}{3} \) and \( \frac{5}{12} \): \( \frac{1}{2}\left(\frac{1}{3}+\frac{5}{12}\right) \) = \( \frac{9}{24} \).
Between \( \frac{5}{12} \) and \( \frac{1}{2} \): \( \frac{1}{2}\left(\frac{5}{12}+\frac{1}{2}\right) \) = \( \frac{11}{24} \).
So three such numbers are \( \frac{5}{12} \), \( \frac{9}{24} \), \( \frac{11}{24} \).

Illustration 1.19. Find 5 rational numbers between \( \frac{-3}{5} \) and \( \frac{1}{4} \)
Answer:
Give them the common denominator 20: \( \frac{-3}{5} \) = \( \frac{-12}{20} \) and \( \frac{1}{4} \) = \( \frac{5}{20} \).
The integers between −12 and 5 are −11, −10, −9, ..., 3, 4, so the numbers \( \frac{-11}{20} \), \( \frac{-10}{20} \), ..., \( \frac{4}{20} \) all lie in between.
Five of them: \( \frac{-11}{20} \), \( \frac{-10}{20} \), \( \frac{-9}{20} \), \( \frac{-8}{20} \), \( \frac{-7}{20} \).

Add to Your Knowledge

1. Division algorithm. \( \frac{\text{Dividend}}{\text{Divisor}} = \text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}} \), which means Dividend = (Divisor × Quotient) + Remainder.

Example. On dividing 4150 by a certain number, the quotient is 55 and the remainder is 25. Find the divisor.
Answer:
Let the divisor be x. Then 4150 = 55 × x + 25, so 55x = 4125.
x = 4125 ÷ 55 = 75.

2. Converting a non-terminating repeating decimal into a fraction. Numerator: take the complete number without the decimal point and subtract the number formed by the non-repeating digits. Denominator: write as many 9s as there are repeating digits, followed by as many 0s as there are non-repeating digits after the decimal point. Example: \( 0.7\overline{37} = \frac{737 - 7}{990} = \frac{730}{990} = \frac{73}{99} \).

Concept Map

Summary

1. Zero is neither a positive nor a negative rational number.
2. Between any two rational numbers there are infinitely many rational numbers.
3. A recurring decimal such as 1.066666... is written with a bar over the repeating digit: \( 1.0\overline{6} \).
4. By using the commutative and associative laws again and again, a sum (or product) of three or more rational numbers is the same in whatever order the numbers are taken. This is the re-arrangement property.
5. Zero has no reciprocal (multiplicative inverse).
6. Division by zero is not defined.
7. Every rational number can be shown as a point on the number line, but not every point of the number line is a rational number.
8. Dividend = Divisor × Quotient + Remainder.