Can you prove this? a x 0 = 0. It's a theorem.?

Question

It's not just because any number multiplied by zero is equal to zero. You should use postulates and axioms.

Answer 1

You know there are 8 axioms in mathematics which we cannot prove

1) if 'a' and 'b' are integers then 'a+b' is an integer and 'a.b' is an integer (Closure law)
2) a + b = b + a and a..b = b.a (cumulative law)
3) a + (b + c) = (a + b) + c (Associative law)
4) a (b + c) = a.b + a.c (Distributive law)
5) a + 0 = a = 0 + a (Additive Identity)
6) a.1 = a = 1.a (Multiplicitive Identity)
7) for all integer 'a' there is a unique integer (-a) such that a + (-a) = 0
8) I can't remember...

So we can prove your requirement using above axioms

0 + 0 = 0 (Obivious)
a ( 0 + 0 ) = a.0 (Multiplying by same factor)
a.0 + a.0 = a.0 (Distributive over addition)
a is an integer and 0 is an integer so a.0 is an integer, hence there exits a unique -(a.0) such that -(a.0) + a.0 = 0

so adding -(a.0) both sides

-(a.0) + (a.0 + a.0) = -(a.0) + (a.0)
(-(a.0) + a.0 )+ a.0 = 0 (associative law)
0 + a.0 = 0
a.0 = 0 (additive identity)

Answer 2

a(0)=a*0=a/0=0
using the properties of a field:
a•0=a•(0+0)
(0 is the additive identity so 0+0=0)
=a•0+a•0
(distributive property a•(b+c)=a•b+a•c)
therefore:
0=a•0 (cancelation property if a+b=a+c, then b=c).
hence proved.

Answer 3

Basically every mathematical operation belongs to a group or "set" meaning
Operation * with elements x is defined such that
1. x1*x2=x3 forms a group such that x1,x2,x3 belong to same group.
2. There exists a identity element x1*I=x1
.....
In your case a * 0 = 0 but in your case 0 itself do not belong to the "group under multiplication" since inverse of zero doesn't exists. So you will have to assume it a x 0 =0. It is not a theorem.

Answer 4

using the properties of a field:
a•0=a•(0+0) (0 is the additive identity so 0+0=0)
=a•0+a•0 (distributive property a•(b+c)=a•b+a•c)
therefore:
0=a•0 (cancelation property if a+b=a+c, then b=c).

Answer 5

Hmm.. If there are zero quantities of a, then a is just an ambiguous quantity... Therefore a does not exist in the material sense. In numerical values, when there are "a" amount of zeroes, when those zeroes are all added together, the result value must be zero.

Answer 6

0/a = 0 [ This to be true and consider already proved. ]

Then,

0 = 0 * a

Hence proved.

Answer 7

come on dude any thing times zero is freakin zero no need 2 prove. sorry i am kind of stupid so wouldent know