Addition Rules

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.

$$a + (b + c) = (a + b) + c \quad {\rm associativity}$$ (3.6)

$$a + b = b + a \quad {\rm commutativity}$$ (3.7)

$$a + 0 = a \quad {\rm identity}$$ (3.8)

$$a + (-a) = 0 \quad {\rm inverse}$$ (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):

$$1 + (3 + 4) = 1 + 7 = 8 = 4 + 4 = (1 + 3) + 4 = 1 + 3 + 4$$ (3.10)

in any possible order or grouping but as we'll see, when we throw in multiplication it can matter a great deal!

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

$$\left\{a \right. = \left.b\right\}$$

$$+\left\{c \right. = \left.d\right\}$$

$$(a + c) = (b + d)$$ (3.11)

In words, the sum of two equations (where we separately sum the expressions on the two sides of the equal signs) is an equation!

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:

$$\left\{ x \right. = \left. y - b \right\}$$

$$+\left\{ b \right. = \left.b\right\}$$

$$x + b = y - b + b$$

$$x + b = y + (- b + b)$$

$$x + b = y$$ (3.12)

or $y = x + b$ (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 $b = b$ 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:

$$\left\{ x \right. = \left. y + b \right\}$$

$$+\left\{ -b \right. = \left. -b \right\}$$

$$x + (-b) = y + b + (-b)$$

$$x - b = y + b - b$$

$$x - b = y + (b - b)$$

$$x + b = y$$ (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:

$$x^2 = y^3 - y + b$$

$$\to x^2 - b = y^3 - y$$ (3.14)

or

$$x^2 = y^3 - y + b$$

$$x^2 = (y^3 - y) + b$$

$$\to x^2 - (y^3 - y) = b$$

$$b = x^2 - y^3 + y$$ (3.15)

Note that by grouping one can move whole sets of symbols at once as long as they are added to one side or the other. We can move terms in any sum from the left to the right side of an equal sign or from right to left equally easily as in either case we are just adding a suitably framed tautology to the original equation!

A final extremely useful rule is to add zero to either side of an equation in a symbolic form such as $0 = a - a$. For example:

$$y = z^2$$

$$y = z^2 + (a - a)$$

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

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.