We are not going to precisely define the addition of numbers - we presume that the reader has at least learned to add real numbers somewhere by the time that they read this, and this skill also suffices to add complex numbers (where one simply adds the real and imaginary parts - all real numbers - separately).
Instead let us reiterate the properties of adding symbols
for arbitrary numbers in whatever system we're working in.
Note that we have introduced the notion of parenthesizing terms - grouping them together in an expression. You should think of parentheses as being instructions to the reader on the order that should be used when evaluating an algebraic expression. The rule is: Do the arithmetic (or algebra) inside parentheses first. If parentheses are nested, do the innermost ones first, then the next innermost, and so on out to the outermost. When all parenthesized expressions are evaluated, go ahead and add up everything else in left-right order (which will no longer matter because of commutativity).
For expressions containing addition only this won't matter
because (for example - try it with any numbers you like):
Another addition rule that is extremely valuable that doesn't quite fit
anywhere else is the following. Given two equations:
This rule is used a lot in algebra. Here are three common forms
based on adding the same thing to both sides of an equation:
Observe the full derivation of this rule. We added the tautology as an equation to another equation, getting an equation, then grouped and cancelled terms. We show all the steps (we usually won't) to emphasize precisely how we can build a new rule based on old ones we already know!
Note that subtraction from both sides is just adding a negative quantity
and doesn't need a separate rule. That is:
When applying this rule to an equation as you try to solve for some
quantity, it is easiest if you just visualize it as a process.
Mentally move any additive term from one side of an equation to
another while changing its sign:
A final extremely useful rule is to add zero to either side
of an equation in a symbolic form such as . For
It turns out that this addition rule is critical to a process called completing the square that we'll use to derive the infamous quadratic formula later on.
To do algebra successfully, one needs to learn all of these rules so completely that one is never ``remembering'' them (as one does with things one has ``memorized'') but so that one knows them. You want to be able to use any of them easily, flipping terms from one side of an equation to another like lightning as easily as you breathe.
That require (more) examples and practice, practice, practice. But really, they are pretty easy to remember because if you think about them at all, they make sense!
Before we go on, since this is mathematics for science we should point out a very important aspect of using algebraic or numerical addition in e.g. physics, or chemistry, or mathematics. It has to do with units.