What forms can units take?

They have stated in my physics book that all units can be made by combining SI base units.

1. Can we raise one unit to the power of another unit? For instance: ${\rm s}^{\rm m}$.
2. How do we know that there aren't more base units?
3. How are we sure that ${\rm m}$ for example is a base unit and is not made up from other units?
• – Anubhav Goel Apr 10 '16 at 9:35

1. No.
• If we could we could raise a unitless quantity to another unit, hence $e^x$ where $x$ is unitful. We have no idea how we would define that except by the old trick of pretending that the power series $\exp(x) = 1 + x + x^2/2 + \dots$ applies equally well when $x$ is unitful -- but that's an obvious type error, hence we just don't know how to do it.
• (With that said, if $f$ is a function which takes a unitful parameter then we can Taylor expand it, as the $(d/dx)^n~f$ derivatives must have consistent units $[[f]]/[[x]]^n.$ The core issue is more that that we don't have any better ideas of how to do $e^x$ beyond naively extending this definition, with its unitless coefficients.)
• Equally importantly, $n$ is not free to be a real number either. Generally unitful expressions only make sense if $n$ is rational.
2. We can always invent more types and we can usually remove base types as well. It's all a set of social conventions amounting to what's useful to us.
• Deleting units: Would you like to keep distance and time as separate concepts? In everyday life that's useful, in relativity it is less so. In relativity we might just decide that the time dimension is best measured in meters as a distance that light goes in the requisite time, deleting the dimension of time for a dimension of space.
• Adding units: On a computer screen, we introduce new length units like px, the size of a pixel, which have no consistency at all with m, the (former) ten-millionth of a quarter of the circumference of the Earth. Some people even prefer to have two different such units, the px horizontal and the px vertical, so that screen area is no longer in px^2 but rather (px h)(px v).
• A unit is a type that we're attaching to a value. Within that unit, we have a "vector space"; then we knit together expressions of different units with certain product-types to restore multiplication and raising to a rational power.
• Hell, there's even units for horizontal length (e.g. furlongs) vs vertical length (e.g. fathoms) that don't really make sense (as useful units anyway) in other directions. – Emilio Pisanty Nov 11 '15 at 0:55

An attempt at a simpler answer:

1. No. Remember that the symbol for a unit does not represent a number/variable in any sense. It is a label we put on a number to say what the number is "counting". So just as $3^\textrm{penguin}$ doesn't mean anything (penguins can be counted to give numbers, but the word "penguin" is a label, not a number) you can't raise $s$ to the power of $m$.

2. Fundamentally we don't. But the base units we have currently defined are sufficient to construct defined units for all measurable quantities we currently know about. But just as the ancient Greeks didn't know about electrical current and so couldn't have defined the Ampere, there may be physical phenomena which we don't know about which will demand the definition of new units once we discover them. On the other hand, as @ChrisDrost points out, we create new (non-fundamental) units all the time.

3. Which units are fundamental and which are derived is a choice. We have currently chosen the meter as a fundamental unit. As a simple example of this choice, you don't need fundamental units for all of length, time and speed. You only need two of them and then you can define the third in terms of the other two. It would have been a valid choice to define fundamental units for speed and time and then define the length unit as the product of the speed unit and the time unit. But the choices that have been made have been made based on large number of very practical considerations. These choices are reviewed constantly and periodically changed. The SI system is likely about to get a major overhaul in the next few years.

I will answer question 3 since I noticed 1 and 2 have been answered nicely.

1. How are we sure that $\rm m$ for example is a base unit and is not made up from other units?

We know that $\rm m$ is a base unit because we defined it as a base unit. Namely, we defined it as the base unit of length.

Is this confusing? Look at it in a different way. You want to know the distance to the moon, but you don't have anything to compare the distance with. Suddenly, you see a stick on the ground and you decide the distance of one end of the stick to the other is what you will compare the distance to the moon with. Your stick is now your unit of distance because you've decided it has length 1 and you will, from now on, say that the distance to something is the amount of sticks you can put in between you and the object. For example, the distance to the moon might now be 1 billion sticks.

Satisfied, you go home and notice that your door is 20 sticks high by 8 sticks wide and that your bathtub is 15 sticks long by 7 sticks wide by 6 sticks deep. You notice you can just multiply the numbers and call the units of surface and volume sticks squared and sticks cubed, since they uniquely define your description. So you now have your units of surface and volume.

So what you have done now is modeled our universe as a vector space of 3 dimensions and everything in our universe is at a point in this vector space denoted by the distance to you in sticks. Obviously you can go further and compare masses by comparing the mass of the stick and in this way add a dimension to your vector space where the unit vector (the unit of mass) is the mass of your stick. You can do this for all kinds of observables, like time and temperature, and whenever you discover a new property that cannot be described through previously defined units you can add a new dimension to the vector space and define a new unit (e.g. as we did with quark charges: charmness $\pm$1, etc.).

You have created a set of coordinates that can uniquely describe things in the universe. We want the observables to be linearly independent of each other, which is necessary otherwise we would be describing things in two different ways and not give any new information. I can define distance in two different ways, but that doesn't add anything useful to the information. I can also say that the set contains velocity, momentum and mass, but one of those is then redundant, since $\vec{p}=m\vec{v}$. A system that does things correctly is the SI, which is tuned to our daily life (note that it does not contain units of relatively new physics).

The fact of the matter is that it is possible to define any new set of coordinates, as long as every coordinate is linearly independent and the set completely spans the known observables. In this new coordinate system, $\rm m$ isn't necessarily a base unit anymore and can be defined using base units of the new system.

This regularly happens in general relativity. There are many coordinate systems because some are more useful than others.

EDIT: small clarifications

An attempt at a REALLY simple answer:

1. Can we raise one unit to the power of another unit?

No.

2. How do we know that there aren't more base units?

A base unit is defined whenever it is needed. If I am dealing with length, I want to define a base unit for length, which is arbitrary. Once the base unit for length is agreed upon (e.g., the meter), everyone knows what unit to work with. If I am dealing with velocity, I need to describe how far something went in a given amount of time, so I need to define the base unit for time, which is the second. For Newton's 2nd law, I need a base unit for mass. For electrostatics, I need a base unit for charge.

3. How are we sure that m for example is a base unit and is not made up from other units?

BY DEFINITION, a base unit is not made up from other units. In other words, a base unit is defined, not derived. Other units are then derived from the defined base units (e.g., Newtons).

3. How are we sure that m for example is a base unit and is not made up from other units?

By reading the definition of a meter, and seeing that it is not simply made by multiplying or dividing other units, although it is related to the second.

The meter is the length of the path travelled by light in vacuum during a time interval of 1/299 792 458 of a second.

SI base units, International Bureau of Weights and Measures (BIPM)

protected by Qmechanic♦Aug 16 '15 at 18:49

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