Should zero be followed by units?

Question

Today at a teachers' seminar, one of the teachers asked for fun whether zero should be followed by units (e.g. 0 metres/second or 0 metre or 0 moles). This question became a hot topic, and some teachers were saying that, yes, it should be while others were saying that it shouldn't be under certain conditions. When I came home I tried to find the answer on the Internet, but I got nothing.

Should zero be followed by units?

EDIT For Reopening: My question is not just about whether there is a dimensional analysis justification for dropping the unit after a zero (as a positive answer to Is 0m dimensionless would imply), but whether and in which cases it is a good idea to do so. That you can in principle replace $0\:\mathrm{m}$ with $0$ doesn't mean that you should do so in all circumstances.

Answer 1

This is actually a really interesting question.

In principle, "zero" doesn't need units. You can think of units as a multiplier - but multiplying zero by anything still leaves you with zero.

However, when you are talking about a physical quantity, it is very reasonable and appropriate to use units, even if the quantity is zero. And you have to use the correct units.

It's important to think about the situations in which it even makes sense to speak of "zero anything" - because the absence of a certain property has different implications in different situations. Think about this statement:

"The photon has zero rest mass" - in this case, there is no need to specify units. The mass is zero - it is simply a property that the photon does not have.

On the other hand, there are times where you are trying to determine whether something is really zero or not. For example, you might want to determine whether the charge of a neutron is truly zero. A careful experiment might conclude that the charge is $0 ± 1.234\cdot 10^{-34} ~\rm{C}$. The units are necessary - because while the number itself is zero, the uncertainty in the number is finite, and has units.

Finally, it is patently wrong to say "the neutron has 0 kg of charge" - which shows that although it is "nominally" the same as saying "the neutron has 0 charge", the units do matter.

Of course, in situations where the scale is arbitrary (that is, where 0 "units" does not correspond to the absence of the property) you always need to use the units. The example given in several of the answers of temperature (°C, K, F) is a good one. In general I believe this can only be true of intrinsic properties (that is, properties that are independent of the quantity of material).

Answer 2

First, as long as the quantity under consideration possesses a unit, yes, because of the importance of the Consistency of Units or Dimensional analysis. Second, in a minor key: in experimental physics, a pure zero is not likely to happen in practice. Therefore, whenever a zero is very close 0.0000000000257, it is important to know the unit: is it in micro or gigaUnit? The factor makes a huge difference. Third: a measurement is often sounder with some notion of uncertainty. Hence, $m=0 \pm 0.001 \;\mathrm{kg}$ tells a lot about the precision one can expect.

To equate, add or multiply quantities, they should be consistent. When one writes $y=ax+b$, the quantities $y$, $ax$ and $b$ should have the same dimension, i.e. the unit of $a$ times the unit of $x$ should be the same as the unit of $b$.

I believe that one should not add a unitless $0$ to a distance, while adding $0$ meter to that distance makes sense. Even if the quantity is $0$ "unit", I believe it still does matter with products, see xkcd: dimensional analysis.

When it comes to applying a more complicated function (a logarithm, an exponential) to a dimensionful number, the discussion is more involved, see for instance Exponential or logarithm of a dimensionful quantity?. Some advocate for instance that a logarithm is dimensionless (from What is the logarithm of a kilometer? Is it a dimensionless number?).

[EDIT] For real-world fans of dimension analysis, Why dimensional analysis matters by UnitFact:

Answer 3

The question cannot be answered generally because it depends on the situation - on what exactly you mean. If you mean "zero mass" then writing $0 \textrm{g}$ or $0 \textrm{kg}$ or something like that is very reasonable. If you mean an abstract, unitless zero from $\mathbb{R}$ - well, that one is unitless and should be written without units.
It strictly depends on what your numerical value wants to express. Numerical values wanting to express some physical quantity which has units should be followed by the appropriate unit, unitless quantities and abstract numbers should not be.

Answer 4

Once again it is logically confusing levels of concepts. By levels I give an example:

The alphabet is one level.

A book written using the alphabet is a meta-level on the alphabet.

A library full of books is a meta-level on both books and based on the alphabet level.

In the case of zero, in the mathematics of integer numbers or real numbers or any mathematical frame, no units are necessary. Mathematically the number zero is completely defined.

Once one models physical quantities one is at a meta level on mathematics: apples, miles, masses... units are necessary to define what is zero and not there to be measured. Zero apples means nothing about miles or masses or ...

For example, a metasyntax is syntax for specifying syntax, metalanguage is a language used to discuss language, meta-data is data about data, and meta-reasoning is reasoning about reasoning.

:)

Answer 5

I think that the answer is yes, if the quantity you are dealing with has units. So if I'm dealing with a mass of $m\,\mathrm{kg}$ and $m$ happens to be $0$, I still should write the unit.

However, if $m=0$, then it does not actually matter what the unit is so long as it has the right dimensions: $0\,\mathrm{kg} = 0\,M_{\odot}$ for instance. So people are often lazy and leave the unit out.

This is rather similar to the common laziness of writing the zero vector as $0$: $\vec{v} = 0$ for instance. Well, that's wrong since $0$ is generally not an element of the vector space but an element of the field over which it is defined. So really you should write $\vec{v} = \vec{0}$, since $\vec{0}$ is an element of the vector space. But people often do not do that, and it is mostly harmless (although I find it annoying).

Answer 6

If you are considering a quantity then there is a difference between:

• a theoretical value of exactly zero (mathematical zero) when one can use any appropriate unit and that being so then why bother with a unit, and
• an experimental value of zero, e.g. 0.000 with the zeroes being significant, then an appropriate unit must be quoted.

Temperature is different because where zero appears on the scale of temperature depends on the chosen unit. So a temperature of $0 \rm K$ is not the same as a temperature of $0^\circ \rm C$.
For a temperature difference of zero one needs to apply what is written in the first paragraph regarding theoretical and experimental values.

Answer 7

Consider this: If $A=B$ and $B=C$, therefore $A=C$.

Let:

• $A$ be voltage drop on the resistor ($[U]=\mathrm V$),
• $C$ be the temperature of the resistor ($[T]= °C$),
• $B$ be zero. - the resistor is unplugged and put in freezer.

Then $$U=0=T\Rightarrow U=T.$$

For two quantities to be equal, the values and units must be equal. For example $1\ \mathrm{km}=1\cdot1000\ \mathrm m=1000\ \mathrm m$, $T(\mathrm{°F}) = T(K)\cdot9/5(\mathrm{°F/K}) - 459.67 (\mathrm{°F})$.

Following this rule, we get that Volts are equal to degrees Celsius, which is nonsense.

Long story short: Units matters!

Another example is temperature: $0\ \mathrm{°F}\neq0\ \mathrm{°C}\neq0\ \mathrm{°N}=0\ \mathrm{°Ré}\neq0\ \mathrm{°De}\neq0\ \mathrm{°Rø}\neq0\ \mathrm K=0\ \mathrm{°R}.$

Answer 8

I have to face this as I write software for pharmacometrics. First there are dimensions, like volume (L, mL) of a compartment as opposed to amount of drug (g, mg, ng, iu, nM) in it (which is different from bodyweight kg). There is time (s, m, h) which is very different from age (d, w, y).

For most dimensions outside of temperature, a constant (like 0) can be assumed to have the same units as anything it is being added to.

Then you get into more complex quantities, such as exponentials, logs, or powers. For example, you might say in a model that volume $V$ in an individual is a function of typical volume $V_0$ but is adjusted for bodyweight $BW$ and a random effect $\eta_V$: $$V = V_0 (BW/70)^k e^{\eta_V}$$ where you intend $V$ and $V_0$ to have units of Liters. Then technically $ln(\eta_V)$ would be dimensionless, as would $(BW/70)^k$ (but not necessarily). From that you can conclude that the $70$ is the same units as bodyweight $BW$.

The point of all this is to help the user by finding inconsistencies in formulas like this, and maybe do some unit conversion, but you do get to a point where they just have to know what they're doing.

Answer 9

If you formalize dimensional analysis, you end up with the set-wise product of a scalar from $\mathbb{R}$ with a free group on with n generators, where n is the number of "base units" you can talk about.

So one of your unit generators might be mass, another distance, another time, etc.

In this structure, addition is only defined when the group portion aligns exactly, and does nothing to them. It adds the scalar.

Multiplication multiplies both the scalar and the units together.

Now, some "units" may be a scalar multiple times some "base unit", but that is ok.

Once you have generated this abstraction, it becomes clear that 0 m/s is a different thing thann 0 kg, but 0 g is the same as 0 kg, and 1000 g equals 1 kg.

While not definiative, a solid abstraction that leads you to treat the zero values differently is a strong reason to do so.

This structure is no longer a field, but that is ok. Not everything is a field.

Answer 10

Golly, in my opinion, when we are told to count it should be by starting from zero whereby no units need apply. Zero would have some meaning before being sent to work. We better state units for zero when it comes up in description of temperature: "temperature" is just a description of the units to follow such as Fahrenheit, Calvin, or absolute. Degrees with no descriptor can be implicit in the special case of minus 40 degrees for either of the drug-store applications. Get it?