Rational Numbers Class 7 Notes: Chapter 9

Introduction to Rational Numbers

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Introduction: Rational Numbers

• A rational number is defined as a number that can be expressed in the form
  \(\begin{array}{l}\frac{p}{q}\end{array} \)
  , where p and q are integers and q≠0.
• In our daily lives, we use some quantities which are not whole numbers but can be expressed in the form of
  \(\begin{array}{l}\frac{p}{q}\end{array} \)
  . Hence we need rational numbers.

Equivalent Rational Numbers

• By multiplying or dividing the numerator and denominator of a rational number by a same non zero integer, we obtain another rational number equivalent to the given rational number. These are called equivalent fractions.
• \(\begin{array}{l}\frac{1}{3}=\frac{1}{3}\times \frac{2}{2}=\frac{2}{6}\end{array} \)

  ∴

  \(\begin{array}{l}\frac{2}{6}\end{array} \)
  and
  \(\begin{array}{l}\frac{1}{3}\end{array} \)
  are equivalent fractions.
• \(\begin{array}{l}\frac{15}{25}=\frac{15\div5}{25\div5}=\frac{3}{5}\end{array} \)
  ∴
  \(\begin{array}{l}\frac{15}{25}\end{array} \)
  and
  \(\begin{array}{l}\frac{3}{5}\end{array} \)
  are equivalent fractions.

Rational Numbers in 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.
• Example: Reduce
  \(\begin{array}{l}\frac{-4}{16}\end{array} \)
  .Here, the H.C.F. of 4 and 16 is 4.
  \(\begin{array}{l}\Rightarrow \frac{-4}{16}=\frac{\frac{-4}{4}}{\frac{16}{4}}=\frac{-4}{16}\end{array} \)
  \(\begin{array}{l}\frac{-1}{4}\end{array} \)
  is the standard form of
  \(\begin{array}{l}\frac{-4}{16}\end{array} \)
  .

To know more about Rational Numbers, visit here.

LCM

• The least common multiple (LCM) of two numbers is the smallest number (≠0) that is a multiple of both.
• Example: LCM of 3 and 4 can be calculated as shown below:
  Multiples of 3: 0, 3, 6, 9, 12,15
  Multiples of 4: 0, 4, 8, 12, 16
  LCM of 3 and 4 is 12.

Rational Numbers Between 2 Rational Numbers

Rational Numbers between Two Rational Numbers

• There are unlimited number(infinite number) of rational numbers between any two rational numbers.
• Example: List some of the rational numbers between −35 and −13.
  Solution: L.C.M. of 5 and 3 is 15.
  ⇒ The given equations can be written as
  \(\begin{array}{l}\frac{-9}{15}\end{array} \)
  and
  \(\begin{array}{l}\frac{-5}{15}\end{array} \)
  .
  ⇒ −615,−715,−815 are the rational numbers between −35 and −13.

Note: These are only few of the rational numbers between −35 and −13. There are infinite number of rational numbers between them. Following the same procedure, many more rational numbers can be inserted between them.

To know more about Rational Numbers Between 2 Rational Numbers, visit here.

Properties of Rational Numbers

Properties of Rational Numbers

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Addition of Rational Numbers

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Subtraction of Rational Numbers

Multiplication and Division of Rational Numbers

Multiplication of Rational Numbers

To know more about Arithmetic Operations on Rational Numbers, visit here.

Negatives and Reciprocals

Negatives and Reciprocals

Additive Inverse of a Rational Number

Representing on a Number Line

Rational Numbers on a Number Line

To know more about Representation of a Rational Numbers on a Number Line, visit here.

Comparison of Rational Numbers