no there is no other symbol
If
0 * 1 = 0
and
0 * 2 = 0
then the general case
0 * x = 0
looking at this general case I can say
x = 1
which is true, or I can also say
x = 2
which is also true.
Now I have the equation
x + x = 1
and now solving for x the only answer is
x = 1/2
is there a different sign for saying that x = 1/2 and no other possible answer?
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Any number multiplied by zero gives zero. You can see here an example of why it isn't possible to divide by zero : it would mean that can be any real. Your way to showing it is not appropriate, I think, since dividing by zero is not allowed. Here is how I would have done it :
Let .
because . Since is different from zero, then the value of this expression is necessarily zero, because the top member of the fraction is null, and therefore this expression is valid for any real .
I avoided the division by zero, but one problem subsists : I did not consider the case when
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You can say that if and only if , then this expression is valid.
From what I'm getting your saying I can't have the equation
0 * x = 0
because that is equivilent to dividing by zero...even though I never did any division.
The reason we are not allowed to divide by zero isn't because it can't be done, its because it gives us every possible answer. Which is pretty useless if your trying to solve an equation for something. With
x / 0 = y
if x > 0 then y = infinity, if x < 0 then y = - infinity, if x = 0 then y is equal to every possible number.
But anyway, even without the divide by zero we can still have an equation such that more than one answer is true.
4^(1/2) = 2
4^(1/2) = -2
Or even
sin(x)=1
x has an infinite number of possible answers.
I was thinking I could write
x + x = 1
x = 1/2 , x not less than 1/2 and x not greater than 1/2.
But thats pretty ridiculous. There must be a sign that means x = y and only y. Or am I using the equals sign wrong. Is it wrong to say that
4^(1/2)=2
???????????????
This is wrong. It doesn't give us any possible answer, it just is undefined in our number system. If you divide by zero, you are not in the traditional number system anymore, and even the most basic stuff (factorization, developing) does not apply anymore.The reason we are not allowed to divide by zero isn't because it can't be done, its because it gives us every possible answer.
Don't complicate stuff. You have :
That is equivalent to .
Factorize with the common factor , it becomes . That is equivalent to , and thus . Not greater, not lesser, it is the only value of that will ever satisfy this equation in our number system.
For example, take the example. It is known that the sinus function is periodic of .sin(x)=1
x has an infinite number of possible answers.
Saying that is correct, but a more precise answer would be that , .
There is no particular sign to say that there is only one answer. In the case of infinitely many answers, we use another parameter, here , to express all the solutions.
We only have one number system, which does not allow division by zero. We did not take time to create another number system in which it is allowed because it is simply not worth it (and it would be difficult, and add to this the problems of having two different number systems).
Anyway, if you are not prepared to accept a fact and are determined to believe that dividing by zero gives every possible value, then there is no point going further in the discussion.