Addition and Multipiication Propertiesof Real Numbers
Axiomsof’Equaligj
For all real numbers a, b, and c :
ReflexiveProperty
Symmetric Property
Transitive Property
a=a
lfa=b,then b=a.
Ifa=band b=c,thena=c.
Substitution Axiom
If a = b; then in any statement involving a we may substitute b for a and obtain another true statement.
Axioms of Addition
Closure
Associative
Additive Identity
Additive Inverses
For all real numbers a and b, a + bis a unique real number.
For all real numbers a, b, and c, (a + b)+ c-= a + (b+ c)
There exists a unique real number 0 (zero) such that a + 0 = 0 + a = a for every
real number a.
For each real number a, there exists a real number —a (the opposite of a)
suchthata+(—a)=(—a)+a=0.
Commutative
For all real numbers aand b, a+b =b+a.
Axioms of Multiplication
Associative
For all real numbers a and b, ab is a unique real number.
For all real numbers a,b, and c, (ab)c = a(bc).
Multiplicative Identity
There exists a unique nonzero real number 1 (one) such that 1-a = a -1= a.
Multiplicative Inverses
For eachnonzero real number a, there ex15tsa real number -—
a
Closure
.
(the reciprocal of a ) such that 51(1) =[-1-)a =1.
a
Commutative
For all real numbers a and b, ab = ba.
The DistributiveAxiom of Multiplication over Addition
For all real numbers a, b, and 0, 61(1)
+ c) = ab + ac.
Definition: Forall real numbersa,b:
a
1