With any two rational numbers a,b we can
define in a unique way

(P1)
If a,b,c are any three real numbers then:
a + ( b + c ) = ( a + b ) + c
Associative Law in Addition
(P2)
If a is any real number then
a + 0 = 0 + a = a
(P3) For every real number a, there is a number a such that
( a ) + a = a + ( a ) = 0
(P4) If a,b are any real numbers, then
a + b = b + a Commutative Law in Addition
(P5) If a,b,c are real numbers then
a ( b c ) = ( a b ) c Associative Law in Multiplication
(P6) If a is any real number then
a 1 = 1 a = a
(P7) For every number there is a number such that
(P8) If a,b are any real numbers, then
a b = b a Commutative Law in Multiplication
(P9) For any numbers a,b,c that belong to the set of real numbers
a ( b + c) = a b + a c Distributive Law
The subtraction and
the division are defined with the help of addition and multiplication
respectively, as below:

(P10) If a=b and c=d then
a + c = b + d and a c = b d
That is we can add and multiply the sides of two equations.
(P11) If a=b then
a + c = b + c and a c = b c
That
is, we can add the same number to both of the sides of an equation.
Also,
we can multiply its side of an equation with the same number.
(P12) If
a + c = b + c then a = b
If a c
= b c and
c is not equal to 0, then a = b.
So, we can delete from both the sides of an equation a common addendum
or the
same non zero factor.
(P13) a
0 = 0
If a
b = 0 then a = 0 or b = 0
(P14) (
1 ) a
=  a
( a )
b =  a
b
( a )
(  b ) = a
b
(P15) 
( a+b ) =  a  b
(P16)