Dimensional Analysis Calculator

What is this?

This is a tool to analyze and simplify units. It can convert any compound unit to pure SI units and simplify it as much as possible. It accepts simple keyboard input, so you can type "kg*m/s^2" and it will interpret it as $\frac{kg \cdot m}{s^{2}}$ . It then converts your input to pure SI units by expanding any derived units you typed in and finding any SI prefixes. It then tries to express the result as simply as possible, so it will convert "kg*m/s^2" to "N".

Features

Accepts all base and derived SI units.
Accepts all SI prefixes.
Accepts some "untypable" characters, like "ohm" → Ω and "micro" → µ
Underlines errors.

Shenanigans and gimmicks

(ste)radians

Technically, rad and sr are defined as $\frac{m}{m}$ and $\frac{m^{2}}{m^{2}}$ respectively. This means that they are dimensionless, but they are still treated as units in this calculator. If they were treated as they are defined, they would be simplified to 1, which would be misleading.

K vs °C

The calculator treats K and °C interchangeably. One case where it is fine is the specific heat capacity:

c = \frac{ΔE}{m \cdot ΔT}

The unit of c is expressed as:

\frac{J}{kg \cdot K} = \frac{J}{kg \cdot °C}

In this case the two units are in fact equal. This is due to how the original equation was written. Notice the delta symbol before the T. It means that the equation takes a difference between two temperatures. And because a difference of one degree celcius is by definition equal to the difference of one kelvin, in this case, both units are interchangeable.

Now take another example, the Boltzmann constant:

k_{B} = \frac{E}{T}

The unit of k_B is expressed as:

\frac{J}{K} \overset{!}{\neq} \frac{J}{°C}

Here the two units are not equal. This is because the equation takes the value of temperature. It is not a difference. And because the conversion between K and °C is linear but offset, the two units are not interchangeable.

The calculator does not know the underlying equation. It only sees the unit. It will convert between the two units blindly, so you should be aware of the underlying physics.

μkg?!?!

The next shenanigan will require a bit of an introduction, so bear with me:

The unit kg is the worst thing there is in the metric system. Think what is compelling in SI? Why did nearly all countries switch to it a few hundred years ago, rejecting other imperial-like systmes? It is the simplicty. Each metric unit has a simple conversion to any other. There are no multipliers like 12 or 5280. Everything is a power of 10. SI has to be simple.

And then there is the kilogram. The only SI unit that includes a prefix. This fact introduced complications to this calculator. The algorithm actually first treats SI as having the gram as the base unit, and then converts it to kilograms. I still feel it is a bit hacky. The resulting shenanigan is that apart from the proper input: "g", "mg", "Tg", "ng" you can also type "mkg", "μkg", "Mkg" and "pkg" respectively and the calculator will treat them equally. It could be fixed easily with a single line of code, but I decided to leave it as a protest.

floating point

The calculator uses floating point numbers, which have a limited range, so the calculator will not handle numbers larger than 10³⁰⁸ or smaller than 10^-308. You wouldn't notice that, but I'm sure you're scrolling up to try it now.

Alternatives

Out of curiosity I checked if there are any other tools online named "dimensional analysis calculator" and they all turn out to be very simple converters or wolfram alpha embeds. The first group lets you compare base or derived units of the same dimension between each other (so they can say that there are 1000 meters in a kilometer) which is incredibly lame for what they claim to be.

The second group of websites are pages with a single low effort article and a wolfram alpha embed. These websites are also lame, because their makers did not do anything special. Wolfram alpha on the other hand has a very interesting feature. You can give it a list of dimensions and it will pick their powers so that they multiply and divide to one. This is very powerful, because you can give it for example acceleration, mass, force and it will come up with F=ma for you.

Comparing wolfram alpha to this site, it can also guess the derived unit given the definition. So you can enter kg*m/s^2 and it will correctly recognize it's a newton. But if you throw anything more complex at it, it will give up. (Example: kg^2 * m^4 / s^7 * A^4 - wolfram doesn't do anything, while this site simplifies it to Ω/F). Another advantage of this site over wolfram is that it has live preview of the input and the result. It is also open-source and free.

The CCC system of measurement

Congrats! You found the button! Now you can convert any SI unit to CCC and back.

The system's name stands for c, calorie, C₄. It is a joke system made by jan Misali that only includes units that begin with the letter C. To cover all units possible to express with SI, the author decided to use the following system, also called "The Seven C's":

Unit	Definition	SI counterpart
c	speed of light in vacuum	$299 792 458 \frac{m}{s}$
cal	calorie	$4.184 J = 4.184 \frac{kg \cdot m^{2}}{s^{2}}$
C₄	the musical note C₄	$220 \sqrt[4]{2} Hz = 220 \sqrt[4]{2} \frac{1}{s}$
°C	degree celcius	$K - 273.15$
C	coulomb	$C = A \cdot s$
cd	candela	$cd$
cent	a hundred particles	$~ 1.66 \cdot 10^{-22} mol$

This idea seemed very fun to me, so I decided to include it in this calculator. There is a caveat though, namely this calculator treats radians and steradians as separate units, and there are no angle measures in CCC.

My solution was to extend The Seven C's into The Nine C's, by adding two more units, names of which come from a circle and a cone:

Unit	Definition	SI counterpart
cir	The angle subtended by a circle at its centre.	$2π rad$
con	The solid angle subtended by a $\frac{1}{8} cir$ cone at its vertex.	$\frac{1}{π (2 - \sqrt{2})} sr$

It is worth noting that originally jan Misali used the roman numeral C for the unit cent, but I decided it could be confused with coulomb on some devices (small display / weird font), so I used the word cent instead. It could also be hard to type, especially on keyboards manufactured after the fall of the Roman Empire.

To read the rest of the article, find an easter egg.
Hint: look down