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Basic Algebra

This page contains central results which are well-known at senior high school level but not always proved. In order that a proof make sense it is important that axioms and other fundamental assumptions are stated explicitly. This is so, in particular, when the proved results seem self-evident.
 

Axioms

 
a+b=b+a
commutative law for addition
a·b=b·a
commutative law for multiplication
(a+b)+c=a+(b+c)
associative law for addition
(a·bc=a·(b·c)
associative law for multiplication
a·(b+c)=a·b+a·c
distributive law
a+0=a
the number 0
a+(-a)=0
opposite number
a·1=a
the number 1
a·(1/a)=1 om a<>0
inverted number
   

Definitions

 
a-b=a+(-b)
defines subtraction
a/b=a·(1/b) if b<>0
defines division
 

Priority

Substitution

Theorems

The nine axioms are exactly those defining a field in abstract algebra. So, they hold for the rational numbers, the real numbers, and the complex numbers separately. Some examples of the implications of the above formulas follow.
 
 

Rational numbers

 

Real numbers

Complex numbers