Simple proof?

Question

prove that for any number x, x•0=0.

Just want to see some of the answers :)

Well, since x•1=x is part of the definition of a field, I don't think I'm going to ask that question. But thanks for insulting me in such a colourful manner, I'm very offended.

Nice Mathematician, just as they teach you in school. And you were able to do so without being rude, also a nice quality.

Answer 1

x*0=x*(0+0)=
x*0+x*0.
The first equality is because 0 is an additive identity;
the second from the distributive law.
Now add the negative of x*0 to both sides:
0=x*0+(-)x*0=
(x*0+x*0)+(-)x*0=
x*0+(x*0+(-)x*0)
=x*0 +0=
x*0.
The first equality is from the definition of a negative;
the second from the previous result; the third from
associativity; the fourth from the definition of a negative;
the fifth from the fact that 0 is an additive identity.

Answer 2

this is a simple question but has a deeper implication. the reason why (usually) X*0=0 is that 0 is the additive identity element for Z, Q , R, and C. N however does not have an additive identity simply because 0 is not a member of N.
therefore, although X*0=0 for our usual Z, Q, R, and C operations, it might not be always so.

Answer 3

x has an additive inverse, -x.

0 = (-x) + x
= (-x) + x*1 [ x = x*1 ]
= (-x) + x*(1+0) [ 1 = 1 + 0 ]
= (-x) + (x*1 + x*0) [ distributive law ]
= (-x) + (x + x*0) [ x*1 = x ]
= ((-x) + x) + x*0 [ Associative rule for addition ]
= 0 + x*0 [ -x + x = 0 ]
= x*0 [ 0+z = z ]

Answer 4

Well, if you were to see this question,
x.0 = x.(1-1) = x.1 - x.1 = x - x = 0
(Unless you were to ask another stupid question to prove x.1=x)
So that's why kids nowadays are taught:
x.0=0.

Answer 5

any number times 0 is going to be 0 simple as that you cant multiple something by nothing.

Answer 6

Simple proof? Oh yes indeed.
Without much fancy ‘footwork’:
0=1/infiniry or x/infinity =0 iff x is finite (negative infinity
Have fun

Answer 7

I'm sure this isn't proper, but:

x*2=x+x
X*1=x
X*0=
(nothing = 0)

Answer 8

let 0=a-b
then x.0 = x(a-b)
=xa - xb
but if a-b=0 then a=b
xa-xb = xa-xa = xb-xb = 0
so x.0 = 0