Expert: Josh - 8/29/2009

Question
Hello:

Why are the units not used with an algebra equation? For example,

Five percent times some number equals one hundred dollars.

The above would be expressed as follows: 0.05x = 100

I do not think it would be expressed as 0.05?x = $100.

Suppose a calculation needs to show unit cancelation?

I thank you for your reply.

Answer
Kenneth,

The general idea behind algebra is to capture mathematical information (read "certain relationships") in the simplest form possible. Units are often considered as non-essential, as far as solving equations are concerned. A minimum description generally involves only numbers. This is deemed both a necessary and sufficient form of representation.

The main point is that the person using algebra should know what they are doing. The algebraic steps or operations are not supposed to tell the whole story. It is not even supposed to indicate the context of the problem. This is why I often suggest that it would be simpler and generally good practice to leave units out of equations.

When you solve equations algebraically, all that is required is that each step must follow logical progression. Formally, the arguments used must obey the axioms of algebra -- [see http://www.math10.com/en/algebra/introduction-to-algebra/algebra-axioms.html for more in-depth information].

Not indicating the context of the problem has its advantages. For example, when you write 0.05x=100, x=2000 whether the "100" represents money, length, weight, temperature or any other physical quantity, the answer has to hold. You do realize that both "x" and the number "100" must have the same units in the equation you wrote?

There is a time and place for everything. In dimension analysis (this is often encountered in chemistry and physics problems), it is appropriate to include units in the calculation to show (as you have indicated) how the units cancel. This is done usually when there are MANY quantities involved and for the sake of consistency. In this case, algebra may be used in an APPLIED setting.

In a more general setting, people normally prefer to leave out the scientific units particularly if only a few quantities are involved. Spelling out each and every single unit defeats the purpose of algebra which is to make things simple and elegant.