Axioms of Real Numbers

Introduction to axioms of real numbers:

The two distinct real numbers are combined by means of addition and multiplication. These real numbers are must satisfy the following axioms. Many number of  axioms are followed by the real numbers. In this article we are going to see about axioms of real numbers with some examples and practice problems.

Types of Axioms

The following Laws are axioms of real number

Associative Addition Law                       : a + (b + c) = (a + b) + c
Associative Multiplication Law               : a * (b * c) = (a * b) * c
Additive identity                                     : a+0 = 0+a = a
Multiplicative identity                             : a *1= 1*a=a
Commutative Addition Law                    : a+b  = b+a
Commutative Multiplication Law            : a * b = b * a
Inverse  Law                                           : a + (-a) = 0
Distributive Law                                      : a * (b + c) = a * b + a * c

Examples for Axioms of Real Numbers:

Let take three real numbers 2 and 4 and 6

Problem1:

Apply the real numbers 2 and 4 and 6 in associative addition and Multiplication Laws

Solution:

As per Associative Addition Law: a + ( b + c ) = ( a + b) + c

Here a = 2 and b = 4 and c = 6

2+ (4+6) = (2+4)+6

2+(10) = (6)+6

12 =12 so these real numbers satisfy the associative addition law.

As per Associative Multiplication Law : a * (b * c) = (a * b) * c

2*(4*6) = (2*4)*6

2*(24) = (8)*6

48 = 48 so these real numbers satisfy the Associative Multiplication Law

Problem 2:

Apply the real number 2 in  Additive identity and Multiplicative identity

Solution:

As per Additive identity : a+0 = 0+a = a

a = 2

2+0 = 0+2 = 2 So the  Additive identity proved.

As per Multiplicative identity : a *1= 1*a=a

a = 2

2*1=1*2 = 2 hence proved.

Problem 3:

Apply the real numbers 2 and 4 in commutative Addition and Multiplication Laws

Solution:

As per commutative  Addition Law : a+b = b+a

a = 2 and b=4

2+4 = 4+2

6 = 6

As per commutative Multiplication Law : a * b = b * a

2*4 = 4*2

8 = 8 hence proved.

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Practice Problems for Axioms of Real Numbers:

Problem 1:

Apply the real numbers 8 and 5 and 9 in Distributive Law

Ans:112

Problem 2:

Apply the real number 9 in Inverse Law

Ans:0