Logic and Proof
When talking about numbers (which we usually are when we use the equal sign), we have a few extra properties of equality that come from arithmetic.
The addition property of equality states that if you have numbers A, B, and C such that A = B, then you also know A + C = B + C. That is, you can add the same number to both sides of the equation. But only if you want to.
Let's say that A = B. Which properties are used in making the statement B + C = A + C?
Well, if we start with A = B, the first thing we did is switch their positions around and write B = A. That's the symmetrical property's doing. Then, we added C to both sides. We can do that according to the addition property. So in total, we used the symmetrical and addition properties.
Similarly, the subtraction property says that if A = B then A – C = B – C, and the multiplication property says that if A = B then A × C = B × C. Finally, the division property says that if A = B and C is nonzero, then A/C = B/C. Basically, as long as you do the same exact thing to both sides of the equation, you should be fine.
But whatever you do, never divide by zero. You may already know this, but dividing by zero is very bad. Many esteemed mathematicians and physicists believe that it will create a black hole in the center of the earth that'll swallow the whole universe and erase everything in existence before you can even reach for your eraser. Avoid it at all costs.
If A = B and B = 4, and C = D and D = 0, can we say that A⁄D = 4⁄C?
At first glance, everything looks hunky-dory. Applying the transitive property allows us to say that A = 4 and since we're given that D = C and the division property allows us to divide both sides by the same number, where's the problem?
Wait…does that say D = 0? Uh oh. Abandon ship! Abort the mission! Reject all statements of the sort because dividing by zero would be a huge, disastrous mistake.
Some people lump the distributive property in with properties of equality, but really it's more a property of addition and subtraction. In any case, we might as well tell you what it is now that we're talking about it.
The distributive property says that if A, B, and C are numbers, then A(B + C) = (A × B) + (A × C). That is, you distribute the A to all the things inside the parentheses.