Why would I ask can "A+B=B" assuming A=0?

Question

Every answer I got to my earlier question assumed that A=0. That's silly. Anyone knows that! I was asking for something more complex than that. Can A+B=B, where A does NOT = 0?

There must be an answer to this. I can't imagine that this world does not provide an answer for this.

Answer 1

There is no solution to A+ B = B when A is not zero unless B is infinite as any number added to infinite is infinite.

Answer 2

To prove this, we will go to the fundamental definitions:
A GROUP (G,+) is a nonempty set G together with a binary operation + on G such that the following conditions hold:
1) Closure: for every a, b in G a+b is in G
2) Associativity: for every a,b,c in G a+(b+c)=(a+b)+c.
3) Existence of an identity: There exists an element e such that a+e=a for all a in G.
4) Existence of an inverse: For every element a in g, there exists (-a) in G such that a+(-a)=e=(-a)+a.

Remark: Consider a+b=c+b
Then (a+b)+(-b)=(c+b)+(-b) (existence of inverse)
a+(b+(-b))=c+(b+(-b)) (associativity)
a+e=c+e (property of inverse)
a=c (property of e)


The real numbers (and even complex numbers) form a group under addition. There for the properties hold in (R,+) (and (C,+)).
For the real (complex) numbers, we define e:= 0. Therefore for every a in C, there exists a (-a) in C such that a+0=a and a+(-a)=0=(-a)+a.

Now, use the remark above with a=A, b=B, and c=0
Then A+B=B
A+B=B+0
so A=0.

So there is no A≠0 such that A+B=B.

Answer 3

I believe that there is an axiom or theorem in algebra that states:
If a+b=b then a=0.
This theorem has a given that a and b are both integers, real numbers or complex numbers.

Infinity is not a number, so whoever stated that is not correct.

As one other person stated, if b is a mathematical expression, like a limit that increases without bound, then a can be any finite number.

Answer 4

There is no solution that this is true with numbers except infinity. If we are talking functions, then we may have some solutions such as sine functions, where the solution occilates between two limits. Example: sin(pi)+sin(2pi)=sin(pi)

BTW, just because you can or can't IMAGINE something does not make it true.

Answer 5

Hell no. If A + B were B den
A = B - B = 0
So if A aien't 0, A + B aien't B.
Why d'you crazy people always come up wit stupid questions?

Answer 6

If B is a real number, no. If B can be a mathematical formula, then it's probably possible.

Answer 7

Logically there is no possible answer but I think B=infinity.

Answer 8

no it cant.
proof:
A + B = B
A + B - B = B - B
A = 0

therefore we cant have another value for a except zero

Answer 9

two condition...
1 either b=infinite
2, a,b is imaginary number....
if real.... its nt possible ..see....
a=b=1....
so a+b=b
1+1=1
bt
2=!1...so fr a real no its nt possible....