A Lesson in Rational Numbers

A Lesson in Rational Numbers

Presenting a lesson in rational numbers for Class VIII students. Study and do the questions that follow.

Abstract

  • Rational numbers are an extension of the number system beyond natural numbers, whole numbers and integers.
  • We need this concept to solve equations like \( 2x = 3 \) and \( 5x + 7 = 0 \).
  • You can visualize rational numbers as fractions in the lowest form.
  • We can do operations of addition, subtraction, multiplication and division on rational numbers in the same way as we do with fractions.
  • You can represent rational numbers on a number line.
  • Between any two given rational numbers, there are countless rational numbers.
  • The idea of mean helps us to find rational numbers between two rational numbers.

2.

  • To define formally, a number that can be expressed in the form p/q, where p and q are integers and q ≠ 0, is called a rational number.
  • A rational number p/q is in the lowest or simplest form if p and q have no common factor other than 1 and q ≠ 0.

3.

  • Rational numbers are closed under the operations of addition, subtraction and multiplication. If a and b are two rational numbers, \( a + b \) is also a rational number. Similarly, \( a – b \) and \( a \times b \) are rational numbers.
  • In this regard, rational numbers are akin to integers which are also closed under addition, subtraction and multiplication.
  • You may recall that whole numbers are closed under addition and multiplication only.
  • Rational numbers are not closed under division because for any rational number a, \( a \div 0 \) is not defined. However, if we exclude zero, then the collection of all other rational numbers is closed under division.

4.

  • The operations of addition and multiplication for rational numbers are (i) commutative and (ii) associative.
  • Rational numbers have these properties in common with whole numbers and integers. They are also commutative and associative for addition and multiplication only.
  • Zero is a rational number but it has no reciprocal as \( \frac{q}{0} \) is not defined, where q is a rational number.
  • 1 and -1 are rational numbers, reciprocals of which are equal to the original rational numbers, i.e. 1 and -1 respectively.
  • The rational number 0 is the additive identity for rational numbers.
  • The rational number 1 is the multiplicative identity for rational numbers.
  • If a, b and c, are rational numbers, \( a(b + c) = ab + ac \), and \( a(b – c) = ab – ac \) . This is the distributive property.
  • The additive inverse of the rational number a/b is –a/b and vice-versa.
  • The reciprocal or multiplicative inverse of the rational number a/b is c/d if \( \frac{a}{b} \times \frac{c}{d} = 1 \).

5.

  • In mathematics, we take a set as a collection of distinct objects satisfying some defining criteria. Obviously, a set may incorporate other sets which in that case are called sub-sets. The larger set may itself be a sub-set of a still larger set. It may also be independent of one or other sets. As such, the set of whole numbers includes sub-set of natural numbers, and itself is a sub-set of integers. The relationship will be clear from the title diagram. We may denote set of natural numbers by \( N \), whole numbers by \( W \), integers by \( Z \), and rational numbers by \( Q \).

Floating Point Numbers

  • 0 is rational, 1 is rational; all integers are rational. What about decimals?
  • These are the floating-point numbers in Computer Science – real numbers with fractions. Example: 8.5, 0.0001, -1234.5678, etc.
  • A floating-point number is rational if it has:
    • a limited number of digits after the decimal point (e.g., 9.1234), or
    • an infinitely repeating digit after the decimal point (e.g., 2.333333…), or
    • an infinitely repeating pattern of numbers after the decimal point (e.g. 7.151515…).
  • So, although 3.14159 is rational 3.14159… is not rational or irrational. The latter is an approximation of \( \pi \). This \( \pi \) is irrational because its decimal representation never ends and never settles into a permanently repeating pattern.
  • You will systematically learn irrational numbers in Class IX. Right now, as you are through a lesson in rational numbers, click below to solve questions. Or, take a trip through the evolution of numbers. Have some fun!

Complex Numbers

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