# Should I always include units at every step?

I've seen some controversy when solving physical equations on whether to put units all the time after I insert a numerical value to a variable with dimensions or to put the final unit at the last equality.

A trivial example;

$F= 5\mathrm{N}, m=3 \textrm{kg}, a=?$

$$F=m a \iff 5\,\mathrm{N}= \left( 3 \,\textrm{kg}\right) a \iff a=\frac{5\,\mathrm{N}}{3 \,\textrm{kg}} =\frac{5}{3} \frac{\mathrm{m}}{\mathrm{s}^2}$$

This feels more consistent when merging algebra in physics, as we can divide the equalities at any step and have $\frac{\mathrm{N}}{\textrm{kg}}=\frac{\mathrm{m}}{\mathrm{s}^2}$ while if we didn't insert units we would have $1=\frac{\mathrm{m}}{\mathrm{s}^2}$. Inspite of this, many physics teachers consult me to only includ the units at the end result.

Additional info; We usually work in SI.

Keeping units in ALL steps leading up to your answer is one of the best ways to avoid silly arithmetic mistakes. I teach high school physics, and when my students neglect to write units as they're working, its much easier to make a silly mistake. Using units allows you to double check that you are only adding or subtracting numbers with like units, similar to how you would only add or subtract like terms in Algebra class. Additionally, if (as in your example above) you are solving for acceleration, you would expect the correct unit of $m/s^2$. If you accidentally divide mass by force (which I see students do sometimes) you would know before you get to you answer that something is not right, because a kg/N is not a unit of acceleration.
I'd also like to point out that if you leave units in your calculations, you can see where the units of your final answer are coming from. Instead, if you omit units during the calculations, you may just be tempted to slap the unit that seems to "fit" on your final answer, without the greater understanding of where it came from.

• +1. While Ihle's comment to work symbolically as much as possible is quite apropos. But telling students to carry units along with the numbers once their calculations reach the number stage is excellent advice for introductory level students. Related to getting dimensions wrong is using inconsistent units. Example: Students far too often get goofy results when asked about a satellite orbiting so many kilometers from the center of a planet. The problem is that they treated those distances and the gravitational constant G (6.674 m^3/kg/s^2) as numbers rather than quantities with units. Oct 5, 2014 at 14:01

Inspite of this, many physics teachers consult me to only includ the units at the end result.

They're wrong. It's always correct to include the units at every step.

However, if you're doing work only for yourself, or only for people who understand the role of units well enough not to need them explicitly written, you can get away without writing units. It's just an example of the general rule that you need to be explicit enough to be clearly understood, but just how explicit that is depends on your audience, not on whatever rules someone else may tell you. When in doubt, it is better to be more explicit, i.e. write the units unless you know your audience will be able to deal without them.

• The statement "it is always correct to include units" does not necessarily imply "it is wrong not to include units" . The advice is mostly not about "audience" but about "avoiding silly mistakes". Once you go to numbers, you lose the consistency check that comes more naturally when you are working in symbols. Oct 6, 2014 at 2:51
• Yes, I intentionally did not imply that it's wrong not to include units, and my second paragraph is meant to explain how it can be considered not wrong to omit units. But I thought the other answers had addressed the importance of units as a consistency check sufficiently well that I didn't feel the need to include it here as well. Oct 6, 2014 at 3:01

I never include the units until the end - but then I never include numbers, either.

I advocate writing the entire analysis in symbols (which implicitly have units, but you don't need to write them down since you "know" (presumably) that $g$ has units of $m/s^2$, for example).

In your case, I would write

$$F = m\cdot a \implies \\ a = \frac{F}{m}\\ \text{substituting values:}\\ a = \frac{5 kg \frac{m}{s^2}}{3 kg} = \frac{5}{3} \frac{m}{s^2}\\ \text{This has units of acceleration as expected.}\\$$

When you have solved the quantity of interest in symbols, you check your units at the end to make sure it still makes sense. If it doesn't, you might check intermediate points.

Only use numbers when you absolutely have to, and not before. Note - when you have addition or subtraction, that's usually a good "intermediate point" to check for silly mistakes. That's where adding apples and oranges is guaranteed to cause trouble (but also it's a problem that ought never to resolve itself before the final line of the analysis so it would show up when you do the unit check at the end).

This is not "high school advice", but is is advice from someone who has used physics professionally for over thirty years... Maybe it's a matter of experience - until you are experienced enough to do this in your head, you should do it on paper.

But honestly I see my children do this (as this is what their teacher tells them) and it makes me cringe.

• Last line: What makes you cringe? Jul 19, 2022 at 14:27
• @Filippo it makes me cringe to see people working a problem with numbers from beginning to end instead of symbols until you have a single expression for the quantity you try to evaluate. Jul 19, 2022 at 14:58