Q&A

# Post History

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posted 12mo ago by celtschk‭  ·  edited 12mo ago by Canina‭

#2: Post edited by Canina‭ · 2021-10-04T13:40:47Z (12 months ago)
• In short: Yes. And there are standard ways to distinguish variables from units.
• Let me explain in detail. In physics, we deal with physical properties of objects and systems. Those quantities can be split down into *quantites*, that is, properties that can be quantified. To quantify basically means to tell how large the quantity is.
• A quantity always has a certain *physical dimension* like length or mass (note that this use of “dimension” is separate from the use in geometry; e.g. space has three geometric dimensions, but all those geometric dimension have the same physical dimension, which may be denoted as “length” or ”distance”).
• Now to describe a quantity, we give it as multiple of a *unit*, which is just a predefined quantity of known size. The most obvious example is the kilogram, which until 2019 literally was the mass of the international kilogram prototype. That is, by definition, an object had a mass of one kilogram if its mass was equal to the mass of that prototype. Now most other units (and now, also the mass) are defined in a more indirect way, but the same principle holds: The unit is just a specific quantity. Note that the SI units are not the only units in use; for example, in astronomy, units like the astronomical unit (the average distance between Earth and Sun) or the parsec (the distance from which the Earth radius is seen in an angle of one arc second) are commonly used. But again, those are simply defined quantities.
• Now when giving the size of some arbitrary quantity, we give it as multiple of an unit. For example, if we say an object has the mass of 2 kilograms ($2\\,\mathrm{kg}$), we say that the mass is twice as large as the kilogram. The numeric factor is sometimes referred to as the *value* of the quantity, but note that the value only makes sense in combniation with a unit, and depends on it. A mass of 2 kilograms is very different from a mass of 2 grams, of 2 atomic mass units, or of 2 solar masses. Rather a mass of 2 kilograms is the same as a mass of 2000 grams. Or as equation: $2\\,\mathrm{kg} = 2000\\,\mathrm{g}$. Note that if we omitted the units here, we'd get the nonsensical equation $2=2000$.
• Now let's come to *constants* and *variables.* Constants are, as their name suggests, assumed to be constant. Some are dimensionless (that is, a pure number without any unit, like the mathematical constant $\pi$ or the fine structure constant $\alpha$), others are actual physical quantities (like the vacuum speed of light $c$, which has the dimension of a velocity). Note that the latter can be used as units, if we desire so; indeed every unit is ultimately a constant; the difference between a unit and another constant is simply that we've decided to give quantities as multiples of that unit. Indeed, the speed of light is quite often used as unit; whenever you give a speed as, say, “half the speed of light”, you are using the speed of light as unit. As formula, you'd write $v=\tfrac{1}{2} c$.
• A variable also denotes a quantity, but one that is not constant. It may depend on the time, or it may be a function of the specific system you are looking at (like the lattice constant of a crystal). Note that if you are looking at only one specific system separately, and the variable is actually fixed by the system (such as the lattice constant if you don't consider varying temperature or pressure), you can treat it as constant in that specific consideration (and possibly even use it as unit).
• So in short, a unit is just a constant that we decided to write other quantities as multiples of, a constant in turn is just a variable that doesn't change in the system under consideration, and a variable is just a symbol denoting some quantity.
• Now on omitting units in intermediary steps (more exactly, on run-on equalities, as the one you had in the post I had commented on).
• A run-on equality is just a chain of equalities that is true at the same time. That is, $a=b=c=d$ is the same as $a=b \text{ and } b=c \text{ and } c=d$. Therefore omitting units in the middle is exactly the same as omitting units at the beginning or end.
• Note that you don't write units on *variables* as those are already describing quantities. You are writing units on *numbers* to turn them into quantities. Remember that units are nothing but constants we decided to use as reference, that is, $m=3\\,\mathrm{kg}$ means, quite literally, the mass $m$ is the same as 3 times the mass of one kilogram. Omitting the $\rm kg$ gives the equation $m=3$ which makes no sense if $m$ is supposed to denote a mass (a mass cannot be equal to the number 3, it can only be equal to three times another mass).
• To make a different example, let's assume you wanted to derive the formula for the kinetic energy of a constantly accelerated object at time $t$. The correct derivation of course is:
• $$E = \frac{1}{2}mv^2 = \frac{1}{2} m(at)^2$$
• Now imagine that someone decided to omit the constant mass factor in the middle, and wrote:
• $$E = \frac{1}{2}v^2 = \frac{1}{2} m(at)^2$$
• I guess you'll immediately recognize that as wrong, as the energy is not $\frac{1}{2}v^2$ but $\frac{1}{2}mv^2$, also where does the $m$ pop up in the last equation again?
• Now omitting the units in the middle is *exactly the same.* You are omitting a constant (remember again, an unit is nothing but a constant quantity) in an equation. Omitting a constant from an equation generally will make the equation wrong.
• Now there is one apparent problem, and that is that the standard letters used to denote certain quantities are the same as the letters used for certain units (which in general are not even of the same dimension). Well, there are several ways one typically deals with that (apart from the fact that context usually makes it clear what is meant anyway).
• First, it is generally a good idea to separate general expressions involving variables from expressions involving explicit values. For example, you'd write all variables on the left hand side of an equation, and all explicit quantities on the right hand side. So e.g. instead of writing $x = 10\\,\mathrm{m}/\mathrm{s}\cdot t$ you might write $x/t = 10\\,\mathrm{m}/\mathrm{s}$.
• Another way to distinguish quantities is to use the fact that units are, by convention, always preceded by numbers. Thus by writing $x = t\cdot 10\\,\mathrm{m}/\mathrm{s}$, it is absolutely clear that $t$ is not an unit, while $\mathrm{m}/\mathrm{s}$ likely is intended to be one.
• Finally, in typeset material (print, online), there is the convention that SI units are always written in upright (“roman”) typeface, while variables are written in italics. Thus $3m$ denotes three times a mass (if $m$ is a mass appearing in the problem at hand), while $3\\, m m$ denotes the distance of 3 meters (note also the more subtle typographical convention of leaving a thin space between the number and the unit). You might have noticed that I've consistently used this convention in my answer.
• In short: Yes. And there are standard ways to distinguish variables from units.
• Let me explain in detail. In physics, we deal with physical properties of objects and systems. Those quantities can be split down into *quantites*, that is, properties that can be quantified. To quantify basically means to tell how large the quantity is.
• A quantity always has a certain *physical dimension* like length or mass (note that this use of “dimension” is separate from the use in geometry; e.g. space has three geometric dimensions, but all those geometric dimension have the same physical dimension, which may be denoted as “length” or ”distance”).
• Now to describe a quantity, we give it as multiple of a *unit*, which is just a predefined quantity of known size. The most obvious example is the kilogram, which until 2019 literally was the mass of the international kilogram prototype. That is, by definition, an object had a mass of one kilogram if its mass was equal to the mass of that prototype. Now most other units (and now, also the mass) are defined in a more indirect way, but the same principle holds: The unit is just a specific quantity. Note that the SI units are not the only units in use; for example, in astronomy, units like the astronomical unit (the average distance between Earth and Sun) or the parsec (the distance from which the Earth radius is seen in an angle of one arc second) are commonly used. But again, those are simply defined quantities.
• Now when giving the size of some arbitrary quantity, we give it as multiple of an unit. For example, if we say an object has the mass of 2 kilograms ($2\\,\mathrm{kg}$), we say that the mass is twice as large as the kilogram. The numeric factor is sometimes referred to as the *value* of the quantity, but note that the value only makes sense in combination with a unit, and depends on it. A mass of 2 kilograms is very different from a mass of 2 grams, of 2 atomic mass units, or of 2 solar masses. Rather a mass of 2 kilograms is the same as a mass of 2000 grams. Or as equation: $2\\,\mathrm{kg} = 2000\\,\mathrm{g}$. Note that if we omitted the units here, we'd get the nonsensical equation $2=2000$.
• Now let's come to *constants* and *variables.* Constants are, as their name suggests, assumed to be constant. Some are dimensionless (that is, a pure number without any unit, like the mathematical constant $\pi$ or the fine structure constant $\alpha$), others are actual physical quantities (like the vacuum speed of light $c$, which has the dimension of a velocity). Note that the latter can be used as units, if we desire so; indeed every unit is ultimately a constant; the difference between a unit and another constant is simply that we've decided to give quantities as multiples of that unit. Indeed, the speed of light is quite often used as a unit; whenever you give a speed as, say, “half the speed of light”, you are using the speed of light as a unit. As formula, you'd write $v=\tfrac{1}{2} c$.
• A variable also denotes a quantity, but one that is not constant. It may depend on the time, or it may be a function of the specific system you are looking at (like the lattice constant of a crystal). Note that if you are looking at only one specific system separately, and the variable is actually fixed by the system (such as the lattice constant if you don't consider varying temperature or pressure), you can treat it as constant in that specific consideration (and possibly even use it as a unit).
• So in short, a unit is just a constant that we decided to write other quantities as multiples of, a constant in turn is just a variable that doesn't change in the system under consideration, and a variable is just a symbol denoting some quantity.
• Now on omitting units in intermediary steps (more exactly, on run-on equalities, as the one you had in the post I had commented on).
• A run-on equality is just a chain of equalities that is true at the same time. That is, $a=b=c=d$ is the same as $a=b \text{ and } b=c \text{ and } c=d$. Therefore omitting units in the middle is exactly the same as omitting units at the beginning or end.
• Note that you don't write units on *variables* as those are already describing quantities. You are writing units on *numbers* to turn them into quantities. Remember that units are nothing but constants we decided to use as reference, that is, $m=3\\,\mathrm{kg}$ means, quite literally, the mass $m$ is the same as 3 times the mass of one kilogram. Omitting the $\rm kg$ gives the equation $m=3$ which makes no sense if $m$ is supposed to denote a mass (a mass cannot be equal to the number 3, it can only be equal to three times another mass).
• To make a different example, let's assume you wanted to derive the formula for the kinetic energy of a constantly accelerated object at time $t$. The correct derivation of course is:
• $$E = \frac{1}{2}mv^2 = \frac{1}{2} m(at)^2$$
• Now imagine that someone decided to omit the constant mass factor in the middle, and wrote:
• $$E = \frac{1}{2}v^2 = \frac{1}{2} m(at)^2$$
• I guess you'll immediately recognize that as wrong, as the energy is not $\frac{1}{2}v^2$ but $\frac{1}{2}mv^2$, also where does the $m$ pop up in the last equation again?
• Omitting the units in the middle is *exactly the same.* You are omitting a constant (remember again, an unit is nothing but a constant quantity) in an equation. Omitting a constant from an equation generally will make the equation wrong.
• Now there is one apparent problem, and that is that the standard letters used to denote certain quantities are the same as the letters used for certain units (which in general are not even of the same dimension). Well, there are several ways one typically deals with that (apart from the fact that context usually makes it clear what is meant anyway).
• First, it is generally a good idea to separate general expressions involving variables from expressions involving explicit values. For example, you'd write all variables on the left hand side of an equation, and all explicit quantities on the right hand side. So e.g. instead of writing $x = 10\\,\mathrm{m}/\mathrm{s}\cdot t$ you might write $x/t = 10\\,\mathrm{m}/\mathrm{s}$.
• Another way to distinguish quantities is to use the fact that units are, by convention, always preceded by numbers. Thus by writing $x = t\cdot 10\\,\mathrm{m}/\mathrm{s}$, it is absolutely clear that $t$ is not an unit, while $\mathrm{m}/\mathrm{s}$ likely is intended to be one.
• Finally, in typeset material (print, online), there is the convention that SI units are always written in upright (“roman”) typeface, while variables are written in italics. Thus $3m$ denotes three times a mass (if $m$ is a mass appearing in the problem at hand), while $3\\, m m$ denotes the distance of 3 meters (note also the more subtle typographical convention of leaving a thin space between the number and the unit). You might have noticed that I've consistently used this convention in my answer. To get this effect in LaTeX/Mathjax, use \mathrm{}.
#1: Initial revision by celtschk‭ · 2021-10-04T12:44:12Z (12 months ago)
In short: Yes. And there are standard ways to distinguish variables from units.

Let me explain in detail. In physics, we deal with physical properties of objects and systems. Those quantities can be split down into *quantites*, that is, properties that can be quantified. To quantify basically means to tell how large the quantity is.

A quantity always has a certain *physical dimension* like length or mass (note that this use of “dimension” is separate from the use in geometry; e.g. space has three geometric dimensions, but all those geometric dimension have the same physical dimension, which may be denoted as “length” or ”distance”).

Now to describe a quantity, we give it as multiple of a *unit*, which is just a predefined quantity of known size. The most obvious example is the kilogram, which until 2019 literally was the mass of the international kilogram prototype. That is, by definition, an object had a mass of one kilogram if its mass was equal to the mass of that prototype. Now most other units (and now, also the mass) are defined in a more indirect way, but the same principle holds: The unit is just a specific quantity. Note that the SI units are not the only units in use; for example, in astronomy, units like the astronomical unit (the average distance between Earth and Sun) or the parsec (the distance from which the Earth radius is seen in an angle of one arc second) are commonly used. But again, those are simply defined quantities.

Now when giving the size of some arbitrary quantity, we give it as multiple of an unit. For example, if we say an object has the mass of 2 kilograms ($2\\,\mathrm{kg}$), we say that the mass is twice as large as the kilogram. The numeric factor is sometimes referred to as the *value* of the quantity, but note that the value only makes sense in combniation with a unit, and depends on it. A mass of 2 kilograms is very different from a mass of 2 grams, of 2 atomic mass units, or of 2 solar masses. Rather a mass of 2 kilograms is the same as a mass of 2000 grams. Or as equation: $2\\,\mathrm{kg} = 2000\\,\mathrm{g}$. Note that if we omitted the units here, we'd get the nonsensical equation $2=2000$.

Now let's come to *constants* and *variables.* Constants are, as their name suggests, assumed to be constant. Some are dimensionless (that is, a pure number without any unit, like the mathematical constant $\pi$ or the fine structure constant $\alpha$), others are actual physical quantities (like the vacuum speed of light $c$, which has the dimension of a velocity). Note that the latter can be used as units, if we desire so; indeed every unit is ultimately a constant; the difference between a unit and another constant is simply that we've decided to give quantities as multiples of that unit. Indeed, the speed of light is quite often used as unit; whenever you give a speed as, say, “half the speed of light”, you are using the speed of light as unit. As formula, you'd write $v=\tfrac{1}{2} c$.

A variable also denotes a quantity, but one that is not constant. It may depend on the time, or it may be a function of the specific system you are looking at (like the lattice constant of a crystal). Note that if you are looking at only one specific system separately, and the variable is actually fixed by the system (such as the lattice constant if you don't consider varying temperature or pressure), you can treat it as constant in that specific consideration (and possibly even use it as unit).

So in short, a unit is just a constant that we decided to write other quantities as multiples of, a constant in turn is just a variable that doesn't change in the system under consideration, and a variable is just a symbol denoting some quantity.

Now on omitting units in intermediary steps (more exactly, on run-on equalities, as the one you had in the post I had commented on).

A run-on equality is just a chain of equalities that is true at the same time. That is, $a=b=c=d$ is the same as $a=b \text{ and } b=c \text{ and } c=d$. Therefore omitting units in the middle is exactly the same as omitting units at the beginning or end.

Note that you don't write units on *variables* as those are already describing quantities. You are writing units on *numbers* to turn them into quantities. Remember that units are nothing but constants we decided to use as reference, that is, $m=3\\,\mathrm{kg}$ means, quite literally, the mass $m$ is the same as 3 times the mass of one kilogram. Omitting the $\rm kg$ gives the equation $m=3$ which makes no sense if $m$ is supposed to denote a mass (a mass cannot be equal to the number 3, it can only be equal to three times another mass).

To make a different example, let's assume you wanted to derive the formula for the kinetic energy of a constantly accelerated object at time $t$. The correct derivation of course is:
$$E = \frac{1}{2}mv^2 = \frac{1}{2} m(at)^2$$
Now imagine that someone decided to omit the constant mass factor in the middle, and wrote:
$$E = \frac{1}{2}v^2 = \frac{1}{2} m(at)^2$$
I guess you'll immediately recognize that as wrong, as the energy is not $\frac{1}{2}v^2$ but $\frac{1}{2}mv^2$, also where does the $m$ pop up in the last equation again?

Now omitting the units in the middle is *exactly the same.* You are omitting a constant (remember again, an unit is nothing but a constant quantity) in an equation. Omitting a constant from an equation generally will make the equation wrong.

Now there is one apparent problem, and that is that the standard letters used to denote certain quantities are the same as the letters used for certain units (which in general are not even of the same dimension). Well, there are several ways one typically deals with that (apart from the fact that context usually makes it clear what is meant anyway).

First, it is generally a good idea to separate general expressions involving variables from expressions involving explicit values. For example, you'd write all variables on the left hand side of an equation, and all explicit quantities on the right hand side. So e.g. instead of writing $x = 10\\,\mathrm{m}/\mathrm{s}\cdot t$ you might write $x/t = 10\\,\mathrm{m}/\mathrm{s}$.

Another way to distinguish quantities is to use the fact that units are, by convention, always preceded by numbers. Thus by writing $x = t\cdot 10\\,\mathrm{m}/\mathrm{s}$, it is absolutely clear that $t$ is not an unit, while $\mathrm{m}/\mathrm{s}$ likely is intended to be one.

Finally, in typeset material (print, online), there is the convention that SI units are always written in upright (“roman”) typeface, while variables are written in italics. Thus $3m$ denotes three times a mass (if $m$ is a mass appearing in the problem at hand), while $3\\,\rm m$ denotes the distance of 3 meters (note also the more subtle typographical convention of leaving a thin space between the number and the unit). You might have noticed that I've consistently used this convention in my answer.