Zero is the
addition identity meaning given any
number a, a + 0 = a.
Zero has the
property that 0 * a = 0. Proof
a = a
a*1 = a
a*(1 + 0) = a
a*1 + a*0 = a*1
a*0 = 0
Also, we must have that zero be
diffferent then the
multiplication identity, 1 (where a*1 = a). Otherwise we would have all numbers
equal to 0. Since a*1 = a and a*0 = a, if 1 = 0, then a = 0 for all a.
More generally, zero is the name given to the
addidtive identity in an
abelian group. Also in any
ring, if
one and zero are the same then the ring only has one
element, namely zero itself (some then
define a
ring as having zero different from
one).