axioms of the real numbers
definition of subtraction
a - b = a + (-b)
identity axiom for multiplication
a(1) = (1)a = a, where 1 is unique
identity axioms
identity axiom for additin, identity axiom for multiplication
the additive inverse of a
(-a)
the additive identity element
0
the multiplicative identity element
1
the multiplicative inverse of a
1/a
the reciprocal of a
1/a
inverse axiom for addition
a + (-a) = (-a) + a =0, where -a is unique
associative axiom for addition
a + (b + c) = (a + b) + c
identity axiom for addition
a + 0 = 0 + a =a, where 0 is unique
commutative axiom for addition
a + b = b + a
closure axiom of addition
a + b = c, where c is a unique real number
reflexive axiom of equality
a = a
inverse axiom for multiplication
a(1/a) = (1/a)a = 1, where 1/a is unique, a ≠ 0
distributive axiom
a(b+c) = ab + ac or (a+b)c = ac + bc
associative axiom for multiplication
a(bc) = (ab)c
definition of division
a/b = a (1/b), b ≠ 0
commutative axiom for multiplication
ab = ba
closure axiom of multiplication
ab = c, where c is a unique real number
associative axioms
associative axiom for addition, associative axiom for multiplication
closure axioms
closure axiom of addition, closure axiom of multiplication
commutative axioms
commutative axiom for addition, commutative axiom for multiplication
transitive property of equality
if a = b and b = c, then a = c
addition property of equality
if a = b and c = d then a + c = b + d
uniqueness of addition
if a = b and c = d then a + c = b + d
multiplication property of equality
if a = b and c = d then ac = bd
uniqueness of multiplication
if a = b and c = d then ac = bd
symmetric property of equality
if a = b then b = a
inverse axioms
inverse axiom for addition, inverse axiom for multiplication
another name for addition property of equality
uniqueness of addition
another name for multiplication property of eqyality
uniqueness of multiplication
