In science, one is actually almost never adding up pure numbers. One is
usually adding up numbers of *things*, or representing *physical
quantities* with numbers. We therefore must carefully note that all of
the beautify algebraic rules above apply *only* when the objects
being added are *the same kind of thing!*

This is the fundamental ``apples and oranges'' rule. If I have six
apples and seven oranges and add them together, I don't have thirteen
apples, and I don't have thirteen oranges. Perhaps I have thirteen
fruit, but if I do I should have written, and thought of, this as adding
6 fruit to 7 fruit. This is even more apparent if I subtract - who
knows that subtracting two oranges from 6 apples even *means*?
Adding two anti-oranges to the six apples?

This means that when we write:

(3.45) |

As we noted above, in most scientific disciplines, we use the same symbols over and over again to stand for different specific kinds of quantities that carry different kinds of units that are either implicit - assumed by convention if no units are given - or explicitly indicated in the problem. In physics, the units are usually the Standard International (SI) unit set, where e.g. a mass symbol such as or is in kilograms, a time symbol , , or is in seconds, lengths like , , or are in meters.

This means that expressions like (the sum of a length,
an area, and a volume) or (the sum of a time, a length, a
mass) are *completely meaningless!*. Requiring consistency of units
is a powerful (and additional) *difference* between ordinary
``math'' algebra and algebra used in application to the real world!

This actually is a great boon to those seeking to solve problems! You
can take it as a given that *physical laws are always dimensionally
consistent*. If you try to formulate and solve a problem using them and
end up with things on two sides of an equal sign with *different
units* or a sum on *one* side of an equals sign containing terms
with different units, you can be absolutely certain that your answer is
*wrong* and furthermore, that *you* made a mistake doing the
algebra! This means that you can go back to the algebra and check and
see where you multiplied where you should have divided or the like, and
possibly solve the problem *correctly*.

Of course this is impossible if you've been ``doing algebra with your
calculator'' by multiplying and adding *numbers* in the problem as
you go instead of using *symbols with implicit units* to solve it
algebraically *first*. Which is why you should almost never, ever
do such a silly thing.

Solving a problem with algebra involves a) formulating the problem in
meaningful symbols with implicit or explicit units; b) solving for the
desired quantity or quantities using algebra; c) *checking the units
of the result* and fixing your work if they are inconsistent; d) *only then* substituting in any given numbers and obtaining a number
answer *that has the right units* because you just checked, didn't
you?

Even if you get the wrong answer, your answer is unlikely to be *crazy* if it has the right units. Answers with inconsistent units are
just plain crazy - they could *never* be right.

- Placing the two sides of any equality into
*almost*any functional or algebraic form as if they are variables of that function - Inequalities