Students can go through AP 7th Class Maths Notes Chapter 8 Rational Numbers to understand and remember the concepts easily.
Class 7 Maths Chapter 8 Notes Rational Numbers
→ Set of Natural numbers are denoted by N and is defined as N = {1, 2, 3, ………..}
→ Natural numbers including ‘0’ are called whole numbers, denoted by W.
W={0, 1, 2, 3, ………….}
→ Integers are denoted by Z (or) I and defined as
Z = {………, -3, -2, -1, 0, 1, 2, 3 }
→ Rational numbers : The numbers which can be expressed in the form of \(\frac{\mathrm{p}}{\mathrm{q}}\), q ≠ 0 where HCF of p and q is ‘1’ are called Rational numbers, denoted by Q and is defined as
Q = {\(\frac{\mathrm{p}}{\mathrm{q}}\), q ≠ 0, p, q ∈ Z,(p, q) = 1}
Ex : \(\frac{4}{5}\), \(\frac{10}{9}\) , \(\frac{-17}{13}\), ………… etc.
→ Rational numbers include integers and fractions.
→ In \(\frac{\mathrm{p}}{\mathrm{q}}\), q ≠ 0, p is called numerator and q is called denominator.
→ Equivalent rational number : A rational number can be written with different numerators and denominator, by multiply the numerator and denominator of a rational number by the same non-zero integer, we obtain another number equivalent to the given rational num ber. This is exactly obtaining equivalent rational number.
Ex : 1) \(\frac{2}{3}\)
Multiply both numerator and denominator by 4.
\(\frac{2}{3}\) \(\frac{4}{4}\) = \(\frac{8}{12}\)
2) \(\frac{-10}{7}\) = \(\frac{-10}{7}\) × \(\frac{3}{3}\) = \(\frac{-30}{21}\)
3) \(\frac{-4}{-5}\) = \(\frac{-4}{-5}\) = \(\frac{4}{5}\) × \(\frac{7}{7}\) = \(\frac{28}{35}\)
→ Positive Rational Number : If both numerator and denominator of a rational number are positive such a rational number is called Positive Rational Number.
Ex : 1) \(\frac{9}{7}\), \(\frac{3}{4}\) ………etc.
→ Negative Rational Number : If the numerator is negative and denominator is positive (or) vice versa in a rational number, then such a rational number is called Negative Rational Number.
Ex : 1) \(\frac{-1}{2}\), \(\frac{-7}{4}\) ………etc.
→ The number ‘0’ is neither a positive nor a negative rational number.
→ If both numerator and denominator of a rational number are negative, then what is to be taken as a positive rational number.
Ex : 1) \(\frac{-3}{-5}\) = \(\frac{3}{5}\)
2) \(\frac{-4}{-7}\), \(\frac{4}{7}\)
→ Rational number on a number tine :
→ Rational numbers in standard form :
In a rational numbers, the denominator should be positive and the highest common factor between numerator and denominator is 1, then such rational numbers are said to be in standard form.
Ex : \(\frac{1}{2}\), \(\frac{7}{12}\), \(\frac{19}{17}\), ………..
→ If a rational number is not in the standard form, then it must be reduced to the standard form.
Ex : 1) \(\frac{-15}{20}\) \(\frac{-15}{20}\) = \(\frac{-5 \times 3}{5 \times 4}\) = \(\frac{-3}{4}\)
2) \(\frac{10}{20}\), \(\frac{10}{20}\) = \(\frac{2 \times 5}{4 \times 5}\) = \(\frac{2}{4}\) = \(\frac{1}{2}\)
→ LCM stands for Least Common Multiple.
→ HCF stands for Highest Common Factor.
→ GCD stands for Greatest Common Divisor.
→ Comparison of Rational number : To compare any two (or) more rational number, we must make the denominators of the two (or) more rational numbers and arrange in ascending order for comparison.
Ex : \(\frac{2}{3}\), \(\frac{3}{2}\), \(\frac{1}{4}\)
\(\frac{2}{3}\) × \(\frac{4}{4}\), \(\frac{3}{2}\) × \(\frac{6}{6}\), \(\frac{1}{4}\) × \(\frac{3}{3}\)
\(\frac{8}{12}\), \(\frac{8}{12}\) \(\frac{3}{12}\)
\(\frac{3}{12}\) < \(\frac{8}{12}\) < \(\frac{18}{12}\) ⇒ \(\frac{1}{4}\) < \(\frac{2}{3}\) < \(\frac{3}{2}\)
→ While comparing two negative rational number, we compare them ignoring their negative signs and then reverse the order.
Ex: \(\frac{-7}{-5}\) and \(\frac{-2}{-3}\)
\(\frac{7}{5}\) and \(\frac{2}{3}\)
\(\frac{7}{5}\) × \(\frac{3}{3}\) and \(\frac{2}{3}\) × \(\frac{5}{5}\) \(\frac{21}{15}\) and \(\frac{10}{15}\)
\(\frac{21}{15}\) > \(\frac{10}{15}\)
\(\frac{7}{5}\) > \(\frac{2}{3}\)
→ If there are common factors in the rational number, first we should reduce them into standard form and compare.
Ex: \(\frac{10}{12}\) and \(\frac{12}{14}\)
\(\frac{10}{12}\) = \(\frac{5}{6}\) and \(\frac{12}{14}\) = \(\frac{6}{7}\)
\(\frac{5}{6}\) × \(\frac{7}{7}\) and \(\frac{6}{7}\) × \(\frac{6}{6}\)
\(\frac{35}{42}\) and \(\frac{36}{42}\)
\(\frac{35}{42}\) > \(\frac{36}{42}\)
\(\frac{10}{12}\) > \(\frac{12}{14}\)
→ There is no integer between any two consecutive integers.
Ex : There is no integer between 2 and 3.
→ There exists infinitely many rational numbers between any two rational numbers.
→ Operations on Rational numbers.
i) Addition : While adding rational num¬ber with same denominators, we add numerators keeping the denominators same
Ex : 1) \(\frac{-10}{5}\) + \(\frac{9}{5}\) = \(\frac{-10+9}{5}\) = \(\frac{-1}{5}\)
2) \(\frac{2}{3}\) and \(\frac{7}{2}\)
\(\frac{2}{3}\) × \(\frac{2}{2}\) + \(\frac{7}{2}\) × \(\frac{3}{3}\)
\(\frac{4}{6}\) + \(\frac{21}{6}\) = \(\frac{4+21}{6}\) + \(\frac{25}{6}\)
Note : Additive inverse of a rational number \(\frac{\mathrm{a}}{\mathrm{~b}}\) is \(\frac{-a}{b}\)
\(\frac{-a}{b}\) = \(\frac{\mathrm{a}}{\mathrm{~b}}\) = 0
ii) Subtraction : While subtracting two rational numbers, we add the additive inverse of the rational number that is being subtracted to the other rational number.
Ex : 1)\(\frac{2}{3}\) – \(\frac{4}{5}\)
\(\frac{2}{3}\) + additive inverse of \(\frac{4}{5}\)
= \(\frac{2}{3}\) – \(\frac{4}{5}\) = \(\frac{2}{3}\) × \(\frac{5}{5}\) – \(\frac{4}{5}\) × \(\frac{3}{3}\)
= \(\frac{10}{15}\) – \(\frac{12}{15}\) = \(\frac{10-12}{15}\) = \(\frac{-2}{15}\)
iii) Multiplication : While multiplying a rational number by a positive integer, we multiply the numerator by that integer, keeping denominator unchanged.
Ex : 1) \(\frac{-2}{3}\) × \(\frac{9}{7}\) = \(\frac{-2}{3}\) × \(\frac{9}{7}\) = \(\frac{-2}{1}\) × \(\frac{3}{7}\) = \(\frac{-6}{7}\)
2) \(\frac{-1}{11}\) × 32 = \(\frac{-32}{11}\)
Note: Reciprocal of rational number \(\frac{\mathrm{p}}{\mathrm{q}}\) is \(\frac{\mathrm{q}}{\mathrm{p}}\).
iv) Division : To divide one rational number by other rational number (nonzero) we multiply the rational number by the reciprocal of the other.
Ex : 1) \(\frac{6}{7}\) ÷ \(\frac{1}{3}\) \(\frac{6}{7}\) × reciprocal of \(\frac{1}{3}\)
\(\frac{6}{7}\) × \(\frac{3}{1}\) = \(\frac{18}{7}\)
2) \(\frac{7}{4}\) ÷ \(\frac{4}{7}\) \(\frac{7}{4}\) × reciprocal of \(\frac{4}{7}\)
\(\frac{7}{4}\) × \(\frac{7}{4}\) = \(\frac{49}{16}\)
3) \(\frac{\mathbf{a}}{\mathbf{b}}\) × \(\frac{\mathbf{b}}{\mathbf{a}}\) = 1
Note: The number ‘0’ has no reciprocal.