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