With any two rational numbers a,b we can
define in a unique way
If a,b,c are any three real numbers then:
a + ( b + c ) = ( a + b ) + c Associative Law in Addition
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
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
same non zero factor.
0 = 0
If a b = 0 then a = 0 or b = 0
-1 ) a
= - a
( -a ) b = - a b
( -a ) ( - b ) = a b
( a+b ) = - a - b