Real Numbers


Real Numbers Definition:

Numbers that can be plotted on a number line are real numbers. The number line shows all positive numbers to the right of zero and negative numbers to the left of zero. In brief, all the numbers which the number line contains are called real numbers. 

For Example:  -2 that can be plotted in a number line. So it is a real number.

Let’s take 2, whose value is equal to 1.414. The decimal Number can be plotted in the number line. So, it is a real number.

Real numbers can broadly be classified into rational numbers and irrational numbers.

Set of Real Numbers:

Real numbers are an ordered set of numbers. The set of real numbers can be represented on a straight line as points. This representation as points on the straight line is one of the most important properties of the real numbers. 

Natural numbers: These are also called "counting numbers". Any integer greater than zero is a Natural number. Natural numbers are counted from 1 to infinity. A special symbol “N” represents the natural number.

E.g. N = {1, 2, 3, 4, 5 and so on}.

Whole numbers: If “0” combines to the natural number, it becomes Whole numbers. These are the set of all positive numbers, including zero, but these do not include any fractional or decimal parts. Whole numbers can never be negative. The only difference between the natural number and the whole number is the addition of “0”.

E.g., W = {0, 1, 2, 3, 4 and so on.}

Rational numbers: Rational numbers are broadly classified into two types.

1) Fractions: Fractions are in the form. Fractions are referred to as non-integers. E.g., They are numbers like (top number- the numerator and bottom number- the denominator). The denominator should not be equal to ‘0’. The numerator and the denominator should be integers, such that, a and b cannot be anything like 2 or π

2) Integers: On the number line, there are many other numbers and not just the positive numbers. These are negative numbers such as, -1, -2, -3, -4 and so on. All the positive natural numbers and all the negative numbers along with “0” when put together, form a set of integers. An integer may be positive, negative, or “0”. There are no fractions or decimals in any part of the numbers. It is represented with the symbol “z”. They are broadly divided into two categories. The first one is negative integers and the second one is zero and positive integers.

Negative integers are like -21, -8, -692.

Positive integers are called whole numbers which start at 0 and go on to infinity. So, whole numbers include all positive integers along with 0, and these positive integers also have a special name called natural numbers which includes all positive integers.

E.g., Z = {1, -1, 0, 101 and -101}.

Irrational numbers: The word 'irrational' means illogical. Irrational numbers can’t be expressed in fractional form. Irrational numbers are also called non-terminating or non-repeating decimal numbers.

E.g. √x, er, π (3.14)

Real Numbers Chart:


Properties of Real Numbers:


The properties can be used to manipulate any algebraic expression. 

The property of real numbers allows:

-> Simplified expressions make them easier to work with 

-> Rewrite the equation so they are easier to solve

The basic properties of real numbers include the following:


1. Commutative property

2. Associative property

3. Distributive property

4. Identity Property

Commutative property

The property states for any real number a and b,

a + b = b + a

In the commutative property of addition, if two real numbers add in any order, the sum will be equal.

E.g. 5 + 2 = 2 + 5 =7

In the commutative property of multiplication, for any real number a and b,

a × b = b × a

 It is very similar to the property of addition. Two real numbers can multiply in any order without changing the result.

E.g. 5 × 3 = 3 × 5 =15

This property applies to all the numbers.

The inverse property for any real number “a”, there is a real number (-a) such that

a + (-a) = (-a) + a =0

So, a is the additive inverse of (-a) and (-a) is the additive inverse of a

Ex-  2 + (-2) = (-2) + 2 = 0

Associative property

Associative property of addition, for any real number a, b and c,

(a + b)+ c = a + (b+ c)

It states that real numbers in addition or multiplication can be added or multiplied accordingly in various ways without changing the result.

E.g. (4x + 2x) + 7x = 4x + (2x + 7x) = 11x

In associative property of multiplication, for any real number a, b and c,

(a ×  b) × c = a × (b × c)

 It says that real numbers in a product can be grouped together in any way and get the same result.

E.g. (4 × 5) × 6 = 4 × (5 × 6) 

= 20 × 6    

          = 4 × 30       

          = 120

Distributive property

The property states that the product of a real number (a) with the resulting sum of two real numbers (b + c) is equal to the sum of products of each part of the real numbers. It is the property of multiplication over addition. First, add the numbers inside the parenthesis (b + c), then distributive property allows us to simplify the expression by multiplying every number b and c inside the parentheses by the multiplier a. This simplifies the expression.

i.e,  a (b + c) = ab + ac

E.g. 3 (10 + 2) = 3(12) = 36

3(10) + 3(2) = 30 + 6 = 36

So, 3 (10 + 2) = 3(10) + 3(2)

Identity Property

The Identity Property is made up of two parts: Additive Identity and Multiplicative Identity.

The additive identity property for any real number a

 a + 0 = a

The number0” is known as the additive identity element of the set of Real Numbers.

E.g. 5 + 0 = 5

The multiplicative identity property for any real number a,

a.1 = 1.a = a

It says that, any number multiplied with 1 gives the number itself. ‘1’ is known as the "multiplicative identity." i.e. -2.1 × 1 = 1 × (-2.1) = -2

E.g. 4.34 × 1 = 4.34

Problem- 1

Write the expression (−25.5) + 30.5 in various ways, by using the commutative property of addition and show that both expressions result in an equal answer.

Solution:

(−25.5) + 30.5 = 5  

30.5 + (−25.5)   = 5

30.5 + (−25.5)   = 5

30.5 – 25.5 = 5

(−25.5) + 30.5 = 30.5 + (−25.5) = 5

Problem-2

Rewrite 17 + 20 + 8.5 – 3.5 in two different ways using associative property of addition. Show that the expressions yield the same answer.

Solution: 17 + 20 + 8.5 – 3.5 = 17 + 20 + (8.5 + (−3.5))

In L.H.S.

17 + 20 + 8.5 – 3.5

=17 + 20 + 8.5 + (−3.5)

= (17 + 20) + 8.5 + (−3.5)

= 37 + 8.5 + (−3.5)

= 45.5 + (−3.5)

= 45.5 – 3.5 = 42

Now in R.H.S. ,

17 + 20 + (8.5 + (−3.5))

=17 + 20 + 5

= 37 + 5

= 42

Answer 17 + 20 + 8.5 – 3.5 = 17 + 20 + (8.5 + (−3.5)) = 42

Problem- 3

Expand the expression by using the distributive property

.

Solution:

Problem- 4

Which property does the following equation represent?

Solution:

The rule for Multiplicative Identity Property is a×1= a

The expression given in the question is: 

Hence the property is Multiplicative Identity.