# Thread: Equals and only equals

1. ## Equals and only equals

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?

2. no there is no other symbol

3. $\displaystyle x + x = 1$

$\displaystyle 2x = 1$

$\displaystyle x = \frac{1}{2}$

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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 $\displaystyle \frac{0}{0}$ 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 $\displaystyle x \in \mathbb{R} - {0}$.

$\displaystyle 0 \times x = 0$ because $\displaystyle 0 = \frac{0}{x}$. Since $\displaystyle x$ 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 $\displaystyle x \neq 0$.

I avoided the division by zero, but one problem subsists : I did not consider the case when $\displaystyle x = 0$

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You can say that if and only if $\displaystyle x = \frac{1}{2}$, then this expression is valid.

4. 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

???????????????

5. $\displaystyle \exists !y \in \mathbb{R}(x=y)$
I believe this will work

edit: said out loud is: there is exactly one real number y such that x=y

6. 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.
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.

Don't complicate stuff. You have :

$\displaystyle x + x = 1$

That is equivalent to $\displaystyle 1x + 1x = 1$.
Factorize with the common factor $\displaystyle x$, it becomes $\displaystyle x(1 + 1) = 1$. That is equivalent to $\displaystyle 2x = 1$, and thus $\displaystyle x = \frac{1}{2}$. Not greater, not lesser, it is the only value of $\displaystyle x$ that will ever satisfy this equation in our number system.

sin(x)=1

x has an infinite number of possible answers.
For example, take the $\displaystyle \sin{(x)}$ example. It is known that the sinus function is periodic of $\displaystyle 2 \pi$.

Saying that $\displaystyle \sin{(x)} = 1$ is correct, but a more precise answer would be that $\displaystyle \sin{(x + 2 k \pi)} = 1$, $\displaystyle k \in \mathbb{Z}$.

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 $\displaystyle k$, to express all the solutions.

7. Originally Posted by Bacterius
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,
What number system allows division by zero? And which ones don't?