Consistency of Units

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:

$$y = ax + b$$ (3.45)

in (say) physics, $y$, $a x$ (as a product) and $b$ had better all have the same units and describe the same kind of physical quantity, or be dimensionless - pure numbers. This puts some obvious limitations on what you can do with algebraic transformation!

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 $m$ or $M$ is in kilograms, a time symbol $t$, $t_0$, or $t_1$ is in seconds, lengths like $x$, $y$, or $r$ are in meters.

This means that expressions like $x + x^2 + x^3$ (the sum of a length, an area, and a volume) or $t + x + m$ (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.