# 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