Rational Numbers

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So far, we have learned about the following set of numbers

Natural Number, N = {1, 2, 3, …}

Whole number, W = {0, 1, 2, 3, …}

Integers, Z = {..,-2,-1,0,1,2, …}

We have also learned about fractions and their applications, so it is important to include them in set of numbers.

Rational number is the set of numbers that includes both integers and fractions.

Q = { p/q : p∈ Z, q∈ Z , q ≠ 0 }

➤The word ‘rational’ arises from the term ‘ratio’ & ‘Q’ comes from the word ‘quotient‘.

➤The decimal numbers can be written as an ordinary fraction and, hence, are rational numbers.

➤The integers can be written as a fraction with denominator 1 and, hence, are rational numbers.

“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.”

Equivalent Rational Numbers

By multiplying/dividing the numerator and denominator of a rational number by the same non zero integer, we obtain another rational number equivalent to the given rational number. This is exactly like obtaining equivalent fractions.

e.g.,

−3/5 = −3×2/5×2 = −6/10

−3/5 = −3×(−2)/5×(−2) = 6/−10

−3/5 = −3×3/5×3 = −9/15

∴ −3/5 = −6/10 = 6/−10 = −9/15 are equivalent rational numbers.

Positive & Negative Rational Numbers

Positive rational numbers – Rational numbers whose both numerator and denominator are positive integers or negative integers (division of two negative integers result in positive integer) are called positive rational numbers.

e.g., 2/7, 3/5

Negative rational numbers – Rational numbers whose either numerator or denominator is negative integer are called negative rational numbers.

e.g., −3/5, 4/−7

Rational Number in a Standard form

➤ 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. e.g., 3/2, 5/7

➤ To reduce the rational number to its standard form, we divide its numerator and denominator by their HCF ignoring the negative sign, if any.

e.g.,

\frac {36}{24} =\frac{ 36\div 12}{24\div 12} = \frac{3}{2}

\frac {15}{-20} = \frac {3}{-4} = \frac {-3}{4}

Rational Numbers on Number Line

➤ If a rational number is proper fraction and

positive then, it lie between 0 & 1, (e.g., 3/5)

negative then, it lie between 0 & -1, (e.g., −3/4)

➤ If a rational number is improper fraction – we should convert it to mixed fraction (integer+ fraction) first then, the rational number lie between the integer and the successive integer. e.g., \frac{8}{5}\, \left(1 \frac{3}{5}\right) lie between 1 & 2, −7/3(−21/3) lie between −2 & −3.

Comparing Rational Numbers

➤ Using number line – Rational number on the right side of other rational number would be greater .

➤ Find equivalent rational numbers having same denominator (LCM) then, we can compare them as like fractions .

➤ To compare two negative rational numbers, we can compare them by ignoring their negative signs and then reversing the order.

e.g., to compare −7/5 and −5/3 , we first compare 7/5 and 5/3 . We get 7/5 < 5/3 and conclude that −7/5 > −5/3 .

➤ Between a negative rational number and a positive rational number, it is obvious that positive rational number is greater.

Rational Numbers between two Rational Numbers

➤ Number of integers between two integers are limited (finite). e.g., between 2 and 7, there are 4 integers.

➤ We can find unlimited number of rational numbers (infinite) between any two rational numbers.

→ Convert the given rational numbers into like fractions, then we can find rational numbers between them.

e.g., find rational numbers between 1/3 & 3/5

We can write the given numbers as,

\frac {1}{3} = \frac {5}{15}\, and\, \frac{3}{5} = \frac{9}{15}

Then, rational numbers between \frac {5}{15}\, and\, \frac {9}{15} are \frac {6}{15}, \frac {7}{15}, \frac {8}{15}

Also, we can write the given numbers as,

\frac {1}{3} = \frac {10}{30}\, and\, \frac{3}{5} = \frac{18}{30}

10/30 < 11/30 < … < 17/30 < 18/30

Now, we can say that, rational numbers between 1/3 & 3/5 are —

\frac {10}{30}, \frac {11}{30}, \cdots , \frac {17}{30}, \frac {18}{30}

So, we can insert as many rational numbers as we want between two rational numbers.

Addition of Rational Nombers

➧ If rational numbers are like fractions, simply add their numerator keeping the denominator same. e.g., 5/12 + 7/12 = (5+7)/12 = 12/12 = 1

➧ If rational numbers are unlike fractions, find the LCM of their denominators and convert the numbers into like fractions and then, add as above. e.g.,

→ 5/6 + 6/5 = 25/30 + 36/30 = (25+36)/30 = 61/30

→ −5/6 + 6/5 = −25/30 + 36/30 = (−25+36)/30 = 11/30

Additive Inverse

Opposite (or Negative) of a given rational number is called its additive inverse because when we add them together, the sum we got is zero.

e.g., 4/7 + (−4/7) = 0

∴ Additive inverse of 4/7 is −4/7 and vice versa.

Subtraction of Rational Numbers

➧ If rational numbers are like fractions, simply subtract their numerator keeping the denominator same.

e.g., 5/12 − 7/12 = (5−7)/12 = −2/12 = −1/6

➧ If rational numbers are unlike fractions, find the LCM of their denominators and convert the numbers into like fractions and then, add as above. e.g.,

\frac{5}{6} − \frac{6}{5} = \frac {25}{30} − \frac{36}{30} = \frac {(25−36)}{30} = −\frac {11}{30}

\frac{5}{6} − \frac{-6}{5} = \frac {25}{30} − \frac{-36}{30} = \frac {(25+36)}{30} = −\frac {61}{30} =2 \frac{1}{30}

Multiplication of Rational Numbers

➧ While multiplying a rational number by a positive integer, we multiply the numerator by that integer, keeping the denominator unchanged.

e.g., 2/3 × (−4) = −8/3 ,

3/5 × 4 = 12/5

➧ We multiply two rational numbers in the following way:

→ Multiply the numerators of the two rational numbers.

→ Multiply the denominators of the two rational numbers.

→ Write the product as

\frac {Result\, of\, Step\, 1}{Result \,of\, Step\, 2}

e.g., 2/3 × −4/5 = (2×−4)/(3×5) = −8/15,

4/3 × 2/−5 = (4×2)/(3×−5) = 8/−15 = −8/15 (in standard form)

Multiplicative Inverse

Inverse (or Reciprocal) of a given rational number is called its multiplicative inverse because when we multiply them together, the product we got is one.

e.g.,

∴ Multiplicative inverse of \frac{4}{7} is \frac {7}{4} and vice versa.

Division of Rational Numbers

To divide a rational number by other rational number, we multiplying the rational number with the reciprocal of the other .

e.g.,\frac{2}{3} \div \frac{4}{3} = \frac{2}{3} \times \frac{3}{4} = \frac {2}{4} = \frac{1}{2}

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