Operations on Real Numbers

Key Concepts

• Define sets and subsets
• Represent real numbers on number line
• Compare and order real numbers
• Perform operations on real numbers
• Perform operations on rational and irrational numbers

Operations on Integers 

• When we add two integers have different signs, subtract and keep the sign of the larger integer. 
• When we add two integers that have the same signs, add the integers and keep the common sign. 
• When two integers have opposite signs, their product/quotient is negative. 
• When two integers have the same signs, their product/quotient is positive. 
• While subtracting integers, change to add the opposite. 

Order of operations 

The order of operations is as follows:  

1. Brackets 
2. Division/Multiplication 
3. Addition/Subtraction 

Example: Following the order of operations, solve

3−(4×5)÷2+

Sol: The order of operations is brackets, division/multiplication (from left to right), addition/subtraction (from left to right). 

3−(4×5)÷2+6 = 3−20÷2+6

= 3−10+6

= −7+6

= −1

Decimal numbers 

Rounding off decimal numbers 

• If the critical digit is 0, 1, 2, 3 or 4, then round down. 

• If the critical digit is 5, 6, 7, 8 or 9, then round up. 

Fraction 

A part of a whole is called a fraction. 

Example: Fraction representing shaded region in 

Types of fractions 

1. Proper fraction: The fraction whose value is less than a whole i.e., numerator is smaller than the denominator. 

2. Improper fraction: A fraction which is more than a whole, i.e., the numerator is larger than the denominator. 

3. Mixed fraction: A whole number and a fraction together. 

Real numbers 

Sets and subsets 

• A set is a well-defined collection of objects. 

An organised arrangement of well-defined objects or elements is called a set. 

A set is represented by a capital letter. 

The elements of a set are represented in curly braces {}. 

Example: If A = {1, 2, 3, 4, 5, 6} 

The elements of set A are 1, 2, 3, 4, 5, 6. 

• A subset is a set of which all the elements are contained in another set. 
• Each unique object that belongs to a set is an element of the set. 

Example: Consider the numbers between 20 and 30.  

Let us name the set of numbers between 20 and 30 as N. 

N = {21, 22, 23, 24, 25, 26, 27, 28, 29} 

Let the subset of prime numbers between 20 and 30 be S 

S = {23, 29} 

Let the subset of odd numbers between 20 and 30 be R 

R = {21, 23, 25, 27, 29} 

All the elements of subsets S and R are the elements of set N. 

Representing real numbers on number line 

We can represent whole numbers on number line. 

We can represent integers on number line. 

We can represent fractions on number line. 

We can represent decimals on number line. 

We can represent all the real numbers on the number line. 

Compare and Order real numbers 

• To compare real numbers, follow the steps: 

Step 1: Find the decimal equivalent for each number. 

Step 2: Then plot the numbers on number line. 

• Example: Compare and order 40/11, √324/36, √10 

Step 1: Find decimal equivalent 

40/11 = 3.63-

√324/36 =18/6 =3

√10 = 3.2

Step 2: Plot them on a number line 

Perform operations on rational numbers 

• To add two rational numbers with different denominators, take the LCM of the denominators and add the numerators. 

a/b + c/d = ad/bd + bc/bd = ad+bc/bd

The sum of two rational numbers is a rational number. 

• To multiply two rational numbers, multiply the numerators and denominators. 

a/b ⋅ c/d = ac/bd

The product of two rational numbers is a rational number. 

Perform operations on rational and irrational numbers 

• To add a rational number and an irrational number: 

a/b + c ≠ pq

The sum of a rational and an irrational number is an irrational number. 

• To multiply a rational number with an irrational number: 

a/b ⋅ c ≠ pq

The product of a rational and an irrational number is an irrational number. 

Exercise

• Determine whether set B is subset of set A if:

Set A = {0,1,2,3,4,5,6}

Set B = {3,5,1}

• Order the real numbers from least to greatest: √200, 14, 41/3
• Identify if the solution is a rational number or irrational number: 4/7 + -1/3
• What is the square root of √144/256?
• Arrange the real numbers in descending order: 2/3, 6.33, √32

Concept Map

What have we learned

• The sum of two rational numbers is a rational number.
• The product of two rational numbers is a rational number.
• The sum of a rational number and an irrational number is an irrational number.
• The product of a rational number and an irrational number is an irrational number.