Real numbers include all rational and irrational numbers. Thus real numbers are integers, fractions and decimals. They even include all positive and negative numbers.

The real number system consists of whole numbers, integers, natural numbers, rational numbers and irrational numbers.

Natural numbers: 1, 2, 3,…
Whole numbers: 0, 1, 2, 3, …
Integers: -1, 0, 1, 2, 3, …
Rational numbers: $\frac{-1}{4}$, -0.5, 2, 41
Irrational numbers : $\sqrt{2}$ , $\pi$, sin 60
Non real numbers: When we consider a number $\sqrt{-25}$ , we may not write it this as 5 or -5 since 52 = 25 and (-5)2 = 25 and therefore $\sqrt{-25}$ is not a real number and that’s the reason $\sqrt{-25}$ is a non real or imaginary number.

Set of all rational and irrational numbers are called as real numbers. Imaginary numbers and infinity are not real numbers. We use real numbers to measure continuous quantities. The decimal numbers with an infinite sequence of digits to the right of the decimal point, are also real numbers.

The set of Natural numbers is {1, 2, 3, .....}
The set of Whole numbers is {0, 1, 2, 3, .....}
The set of Integers are {....-2, -1, 0, 1, 2,...}
The set of Rational numbers is { all numbers of the form $\frac{p}{q}$ are a set of rational numbers. Where p, q are integers and q $\neq$ 0}
Irrational numbers are non - repeating and non - terminating decimals, that cannot be written in the form of a quotient of two integers.

Real Number Definition: Rational numbers and Irrational numbers combined together are known as Real numbers. Real numbers include terminating as well as non terminating repeating and non repeating decimals. The Set of Real numbers is a Super Set of many Sets like a set of Rational numbers, a set of Irrational numbers, a set of Natural numbers, a set of Whole numbers, a set of Integers etc. It can be represented by a Venn diagram as follows:


Real Numbers

Is zero a real number ?


Yes, zero is a real number, but it is not a counting number. It is an integer, a real number and a rational number.

There are five properties of real numbers:

1) Commutative Property - It is applicable for both addition and multiplication. It means we can add or multiply two or more numbers in any order.

So, 5+4 = 9 = 4+5 , and 4x5 = 20 = 5x4

2) Associative property - It is applicable for both addition and multiplication. It means we can group two or more numbers in any way during addition and multiplication.

Example (4+3)+2 = 4+(3+2)= 9 and (4x3)x2= 4x(3x2)= 24

3) Distributive property - It is applicable when multiplication and addition both are involved. If we have to multiply a term in parenthesis, we have to distribute the multiplication. The name is actually distributive property of multiplication over addition.

Example:- 4x(3+5) =4x3+4x5 =32

4) Density Property - It means we can always find a real number between any two real numbers. Say between 3.4 and 3.5 we can find numbers like 3.41, 3.42,-and 3.411, 3.412, 4.4111, 3.41112 etc.

5) Identity property - For addition it means zero added to a number is the number itself and 1 multiplied by a number is the number itself. So,4+0=4 and 5x1=5

Real Number Chart


Real Numbers Chart

The following steps for adding real numbers will help us find the sum between two real numbers.

Let us consider the sum of(-7 + 3). We can interpret the expression as the meaning of starting at the origin and then move 7 units in the negative direction and then add 3 units in the positive direction.


Step 1: Start at zero and move 7 units in the negative direction on the number line.
Step 2: After reaching negative 7 on the number line we need to count 3 in the positive direction.
Step 3: Reach the point interpreted as the final answer on the number line.

Adding Real Numbers

Examples on Adding Real Numbers


Given below is an example based on adding real numbers

Add $\frac{8}{12}$ + $\frac{6}{12}$

Solution: Given $\frac{8}{12}$ + $\frac{6}{12}$

Step 1: The denominators are the same go to step 2.
Step 2: Add the numerators $\frac{8}{12}$ + $\frac{6}{12}$ = $\frac{8+6}{12}$ = $\frac{14}{12}$
Step 3: Simplify the fraction $\frac{14}{12}$ = $\frac{7}{6}$

  • Closure : If a and b are two Real Numbers, the a+b or b+a is also a Real Number.
  • Commutativity : For two Real Numbers a and b, a+b=b+a.
  • Associativity : For three Real Numbers a, b and c, a+(b+c) = (a+b)+c.
  • Additive Identity : Since a+0 = a = 0+a , therefore 0 is the Identity element for Addition.
  • Distributivity : For three Real Numbers a, b and c, a×(b+c) = (a×b)+(a×c).
  • Density : Between two Real numbers a and b, there always lies another Real number x.
    such that a $\leq$ x $\leq$ b

The following Steps for subtracting real numbers will help us in finding the difference between two real numbers.

Let us consider the sum of two negative numbers like (-3 - 5)
We can interpret the expression as the meaning of starting at origin zero and then move 3 units in the negative direction and then move another 5 units in the same negative direction on the number line.

Step 1: Start at zero and move 3 units in the negative direction on the number line (-3).
Step 2: After reaching negative 3 on the number line we have to add another 5 units in the negative direction (-3) + (-5).
Step 3: Reach the point interpreted as the final answer (-8) on the number line.

Subtracting Real Numbers

Examples on Subtracting Real Numbers


Given below are examples based on subtracting real numbers

Subtract $\frac{7}{9}$ - $\frac{4}{9}$

Solution: Given $\frac{7}{9}$ - $\frac{4}{9}$

Step 1: The denominators are the same so go to step 2
Step 2: Subtract the numerators $\frac{7}{9}$ - $\frac{4}{9}$ = $\frac{7-4}{9}$ = $\frac{3}{9}$
Step 3: Simplify fraction $\frac{3}{9}$ = $\frac{1}{3}$

Division by a number is the same as multiplication by its reciprocal and for this every problem can be written as a multiplication problem but after replacing the real numbers with their reciprocals.
If ‘a’ and ‘b’ represent any two real numbers and b ? 0 then the expression is always
a ÷ b = $\frac{a}{b}$ = a ($\frac{1}{b}$)

While doing the operation, we have to keep in mind the following product rules.

(Positive) x (Positive) = Positive
(Positive) x (Negative) = Negative
(Negative) x (Negative) = Positive

Examples on Dividing Real Numbers


Let us try one example based on the division of real numbers.

1. Divide 15 by 3 (15 ÷ 3)

Solution: Given (15 ÷ 3)

Step 1: Express in $\frac{a}{b}$ form
Step 2: Change the expression into multiplication [ a ($\frac{1}{b}$)]
Step 3: Write the factor as the final answer.

$\frac{15}{3}$ = 15 ($\frac{1}{3}$) = 5

2. Divide 18 by (-3)

Solution: Given 18 by (-3)

Step 1: Express in $\frac{a}{b}$ form
Step 2: Change the expression into multiplication [18 ($\frac{1}{-3}$)]
Step 3: Write the factor as the final answer keeping in mind the product rules

$\frac{18}{(-3)}$ = 18 ($\frac{1}{-3}$) = -6

  • Closure : If a and b are two Real Numbers, the axb or bxa is also a Real Number.

  • Commutativity : For two Real Numbers a and b, a x b = b x a.

  • Associativity : For three Real Numbers a, b and c, a×(b×c) = (a×b)×c,

  • Multiplicative Identity : Since a × 1 = a =1 × a, therefore 1 is the identity element for Multiplication.

The following examples help us to understand how to perform various operations on real numbers.

1) Is $\frac{1000}{0}$ a real number ?

Solution : It is not a real number as 1000 divided by zero is not defined.

2) Some boys and girls are travelling in a bus for a picnic, the ratio of boys and girls in the bus is 2:4 . 20 boys are present in bus, calculate how many girls are travelling in bus.

Solution:

Step 1: Lets assume girls travelling in bus is x .
Write the ratio in the form of a fraction

Girls = $\frac{2}{4}$ = $\frac{X}{20}$

Step 2: Solve for the equation

Cross multiplying the above equation turns into 2 × 20 = 4 × (X)
40 = 4x
Divide both the sides by 4
$\frac{40}{4}$ = $\frac{4X}{4}$

X = 10
Therefore number of girls travelling in bus = 10 .


3. Add $\frac{5}{9}$ + $\frac{4}{9}$

Solution: Given $\frac{5}{9}$ + $\frac{4}{9}$

Step 1: The denominators are the same so go to step 2.

Step 2: Add the numerator $\frac{5}{9}$ + $\frac{4}{9}$ = $\frac{9}{9}$

Step 3: Simplify fraction $\frac{9}{9}$ = 1


4. Subtract $\frac{6}{8}$ - $\frac{2}{8}$

Solution: Given $\frac{6}{8}$ - $\frac{2}{8}$

Step 1: The denominators are the same so go to step 2.

Step 2: Subtract the numerator $\frac{6}{8}$ - $\frac{2}{8}$

Step 3: Simplify fraction $\frac{4}{8}$ = $\frac{1}{2}$


5. Divide $\frac{\frac{1}{2}}{\frac{4}{2}}$

Solution: Given $\frac{\frac{1}{2}}{\frac{4}{2}}$

Step 1: Invert the fraction

$\frac{4}{2}$ will turn in to $\frac{2}{4}$

Step 2: Multiply the first fraction by the reciprocal

$\frac{1}{2}$ × $\frac{2}{4}$ = $\frac{1\times2}{2\times4}$ = $\frac{2}{8}$

Step 3: Simplify the fraction

$\frac{2}{8}$ = $\frac{1}{4}$