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.

(3.6) | |||

(3.7) | |||

(3.8) | |||

(3.9) |

are all ways (however trivial) of changing one side or the other of an equation.

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):

(3.10) |

Another addition rule that is extremely valuable that doesn't quite fit
anywhere else is the following. Given two equations:

(3.11) |

In words, the sum of two equations (where we separately sum the

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:

(3.12) |

or (following a common but unnecessary convention of putting ``the answer'' on the left and the formula for obtaining it on the right).

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:

(3.13) |

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*:

(3.14) |

or

(3.15) |

Note that by grouping one can move

A final extremely useful rule is to add *zero* to either side
of an equation in a *symbolic form* such as . For
example:

This rule initially looks a bit silly. Why do we need it? We're adding something arbitrary to an equation that

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*.