Rational NumbersMathematics- Grade 7

Module Objectives

• Know the meaning, standard form and equivalents of rational numbers,

• Develop the skill of representing the rational numbers on a number line,

• Know the comparison of rational numbers, finding rational numbers between any two rational numbers,

• Know the operations on rational numbers,• Know the method of writing rational numbers in decimal

form• Know the multiplication, division operation of decimal

numbers and solving problems based on them.• Understand the method of converting measure units.

IntroductionNatural numbers N={1,2,3,4…}Whole numbers W={0,1,2,3,…}Integers Z={…,-4,-3,-2,-1,0,1,2,3,4…}

Rational Numbers are of the form where and a,b are integers .

Examples:

Here are a few videos to help explain the topic:•Introduction video 1•Introduction video 2

We can write all whole numbers, natural number and integers as integers.Example: 1.5 is a rational number because 1.5 = 3/2 (it can be written as a fraction)

In a rational number ,

•if both a and b are positive or both negative integers, then it is a positive rational number.

Example:

•if (a or b) any one is a negativeinteger, then it is negative rational number.

Example:

Standard form(simplest form) of a rational number

A rational number is said to be in its standard form if its numerator and denominator have no common factor other than 1, and its denominator is a positive integer.

•Consider Dividing both numerator and denominator by 2 we get the standard form

•Similarly standard form of (on dividing by 5)

•Standard form of

Equivalent rational numbersEquivalent rational numbers can be obtained by multiplying or dividing both numerator and denominator of a rational number by the same non-zero integer.

•Example: Write 4 equivalent rational numbers for

•Example: Write 4 rational numbers equivalent to

Representing rational numbers on a number line

Divide the unit length on the number line into the number of parts as in the denominator of the rational number. Then mark the number of parts as in the numerator of the rational number. Here’s a Link to a Video on the topic.Example: Represent 3/5 on a number line

Example: Represent -7/4 on a number line

Comparison of rational numbers

If the denominator of two fractions are same, then the fraction with greater numerator is greatest; the same goes for rational numbers.

Examples:

To compare rational numbers having different denominators, convert them using LCM to have the same denominator and then compare them as compared in fractions.

Example: Between 2/3 and 3/5, which is greater?

Example: Between -7/4 and -5/3, which is greater?

Rational Numbers between two rational numbers

We can easily find integers between any two integers. Example: Integers between -2 and 4 are -1,0,1,2,3. There are no integers between 5 and 6 but there are innumerable rational numbers between 5 and 6. To find some of them:

To find rational numbers between two rational numbers having different denominators:

Operations on rational numbers-Addition

If denominator is same, then write the same denominator and add the numerator and write the sum as numerator. If the denominators are different, take the LCM of the denominators, convert them to have the same denominator and then add.

Additive Inverse of a rational number

Additive inverse of 3 is -3Additive inverse of a rational number is the same number having opposite sign in it’s numerator.Additive inverse of

Note: The sum of a number and its additive inverse is ‘0’(zero) : a+ (-a) = a-a = 0

Subtraction of rational numbers

Multiplication of Rational numbers

Multiplicative inverse(Reciprocal) of rational numbers

Interchange the numbers in the numerator and denominator of the rational number.

Note: No change in the sign while writing multiplicative inverse. The product of a number and its reciprocal is always 1.

Division of rational numbers

Multiply the dividend by the reciprocal of the divisor.

Writing rational numbers as decimalsDecimal numbers have 2 parts. For example, in 61.35, 61 is the integer part and 35 is the decimal part.

There are 2 types of decimals:

Examples of converting rational numbers to decimals

Writing decimals as rational numbers

Write the numbers as it is without decimal point as numerator. Then write 1 in the denominator followed by zeroes equal to number of digits in decimal parts.

Multiplication of decimal numbersFind the product of the numbers without taking the decimal point for consideration. Later, put the decimal point in that product by leaving the digits to right side equal to the number of decimal places both in multiplicand and multiplier. Suppose there is no sufficient number of digits in the product write zeroes instead of that.

Dividing decimal numbers

To divide decimal numbers by integers

To divide decimal numbers by decimals

Unit conversionQuantity of measurement gives numerical value. Quantities can be compared by units. A few units of measurement have been accepted internationally. There are called Standard Units. International basic unit of length is metre (m)International basic unit of mass is kilogram(kg)