Properties of equality

Can anyone give me an explanation of why I'm not allowed to add an arbitrary number to both sides of any false equation? What about an unproven equation? If x!=y, then adding a random number will never change that, right? Could some one show me a counter example, where adding the same number to both sides will change whether it's equal or not? I understand that with multiplication and division, if I multiply both sides by zero, the two sides will be equal, but I don't understand why I can't add arbitrary numbers to both sides of an unproven identity. Thanks.

Let x, y, z be real numbers. Then:

x != y . . <=> . . x + z != y + z

(ignore the dots - I only used them to space it out a bit)

What makes you think otherwise?

I am confused with where the confusion is here. As Starofale put it, adding z to both sides will not change the fact that the statement is false. The same goes for a true statement, so I'm not sure what the significance of the statement being false is.

When proving if a statement is true or not, you're not allowed to add something to both sides, so I want to know why not.

Legilimens wrote: When proving if a statement is true or not, you're not allowed to add something to both sides, so I want to know why not.

The reason is that it is an unneeded step. You do not need to add anything to the sides to make the proof because that would just be an additional step in the wrong direction. Did your math teacher not teach this in class?

Suppose you had an (unproven) equation like cos(4)+5=cos(5)+3 that you wanted to prove false. You're not allowed to subtract the same thing from both sides because this isn't an equality. It also isn't a wasted step if I were to subtract 3 from both sides, because it simplifies the expressions. I'm aware that this is a really basic equation, but suppose that it was a lot more complex, and that just looking at it won't tell you if it's false or not.

If looking at the equation does not tell you if it is false or not, you have to calculate the equation out to get your answer. subtracting extra steps does nothing at all. just calculate the answer!

cos(4)+5 = cos(3)+2 0.9975640502598242476131626806443+5 = 0.99862953475457387378449205843944+2 5.9975… != 2.9986…

AldarHawk wrote: If looking at the equation does not tell you if it is false or not, you have to calculate the equation out to get your answer. subtracting extra steps does nothing at all. just calculate the answer!

cos(4)+5 = cos(3)+2 0.9975640502598242476131626806443+5 = 0.99862953475457387378449205843944+2 5.9975… != 2.9986…

Just no.

What can I say. I have not done that math in forever :P and I did it in my head LOL

Ohh well. I am sure if I actually thought it over (which I know spy is gonna have a field day on this) I would get it right :P

Legilimens wrote: Suppose you had an (unproven) equation like cos(4)+5=cos(5)+3 that you wanted to prove false. You're not allowed to subtract the same thing from both sides because this isn't an equality.

So your reason for not believing you can add the same number to both sides is that you can't be sure if the two expressions are equal or not?

Say we need to show whether x is equal to y or not. We have three cases (for real numbers):

1. x < y
2. x = y
3. x > y

In all three cases you are allowed to add a number to both sides and the equality/inequality will remain unchanged.

Yes, I believe that I should be allowed to add numbers to both sides. However, if you ask a math teacher, they will tell you that you aren't allowed to do that. Unfortunately, my teacher wasn't able to give an explanation of why not.