Editing 3038: Uncanceled Units

{{comic
| number    = 3038
| date      = January 15, 2025
| title     = Uncanceled Units
| image     = uncanceled_units_2x.png
| imagesize = 323x355px
| noexpand  = true
| titletext = Speed limit c arcminutes^2 per steradian
}}

==Explanation==
{{incomplete|Created by THE HUBBLE PARAMETER, WHICH I HAD INCORRECTLY REFERRED TO AS THE PLANCK CONSTANT (PLS FORGIVE THIS BOT) - Please continue to explain the joke and possible interpretations. Do NOT delete this tag too soon.}}

Another of [[Randall]]'s [[:Category:Pet Peeves|pet peeves]], this comic expresses disapproval of units that could be mathematically simplified (in other words, the 'uncanceled' of the title refers to mathematical cancellation, not cancellation by a body, like how SI supplanted the CGS system with MKS).

[[White Hat]] is presenting a refrigerator to [[Cueball]], saying it uses 3 kWh per day. This is a common and useful way to report power usage. But mathematically, the units can be simplified because there are two time units that cancel each other:

: 3 kWh per day = 3 kW * 1 hour / 1 day = 3 kW * (1 hour / 24 hours) = 0.125 kW = 125 W

This gives the average power usage in {{w|watts}} (a unit for the rate of energy transfer, equal to 1 {{w|joule}} per second). Alternatively, one could convert it to 3 kWh * 3600 kJ / kWh = 10800 kJ / day, as many refrigerators use very different amounts of power throughout a day.

The reason people would use "kWh / day" without simplifying the unit is that kWh is a commonly used unit for energy, and it's often viewed as a base unit even though it's composite (1 kWh is the amount of energy consumed by one kilowatt of power usage over one hour, and is equal to 1 kJ/s * 1 h = 3600 kJ). It's the unit in which energy consumption is typically reported and in which bills are calculated, so it's more familiar to the average consumer than kJ and megajoule (MJ), and giving the power usage in kWh / day makes it easier for the consumer to understand how much money it will cost them to run per day. But mathematically, "kWh / day" is inelegant, because it uses power (which is already a measure of energy per time) multiplied by a time unit then divided by another time unit. This gets increasingly ridiculous if you reduce all the way down to the seven base quantities, resulting in: mass times length squared over time squared, per time, times time, per time.

Cueball (probably representing Randall) sardonically wonders whether the refrigerator would fit in his kitchen, since the ceiling is only 50 gallons per square foot high. This is clearly an abnormal and unhelpful way of reporting height. This unit turns a normal measurement of height (feet and inches in the US; meters and centimeters most other places) into a weird collection of uncancelled units. Gallons can be transformed to cubic feet (1 US gal = 231 in<sup>3</sup>), which can be divided by the square feet, yielding a ceiling height of around 6 feet 8 inches, or 203.7 cm. (Using imperial gallons [1 UK gal ≈ 277.42 in<sup>3</sup><!--exactly equal to 4.54609 L-->], the height is roughly 8 feet, or approximately 244.7 cm.) This is intended to lampoon the use of both non-metric and uncancelled units by showing how odd things become if they're generally used.

The exact ceiling height in feet, assuming the US gallon is used, can be calculated as:

: 50 gallons per square foot = 50 gal * 231 in<sup>3</sup>/gal * (1/12 ft/in)<sup>3</sup> / 1 ft<sup>2</sup> = 6.68402777... ft

''[[what if? (blog)|what if?]]'': [https://what-if.xkcd.com/11/ Droppings] also covers strange instances of unit cancellation, including a measure of volume per distance converted to area; similar to Cueball's measure of volume per area representing a distance (the height of his ceiling).<!-- This may not be relevant enough to keep -->

A common source of unit drama occurs between lay people who are looking for every day practicality and science/engineering types who are inclined towards formalized mathematical operations. For example U.S. customary units which support many divisibility rules (1 foot = 12 inches; 1 inch = 72 points = 1440 twips; 3 feet = 1 yard; 2 yards = 1 fathom; 22 yards = 1 chain; 10 chains = 1 furlong; 1 mile = 5280 feet; 1 league = 3 miles) versus metric units which prioritize base 10 scales. In this case, telling the average customer the energy use in joules per day or average consumption in watts would require them to perform more complicated conversions to get to the figure they actually care about — the actual cost per day. White Hat could just give this cost figure directly, but does not know what every customer pays for electricity (an explicit yearly cost estimate would be included on the government required energy efficiency label).

In the title text, a speed limit is given as c arcminutes<sup>2</sup> per steradian, where c is presumably the speed of light in vacuum — 2.998×10<sup>8</sup> m/s (meters per second) or 186282 mi/s (miles per second). A steradian (sr) is the SI unit for solid angle, subtended by a section of a sphere (like a radian is a unit of angle subtended by a section of a circle). A square arcminute is also a unit of solid angle, equivalent to a section of a sphere of 1/60 of a degree by 1/60 of a degree. There are ((1/60)*(π/180))<sup>2</sup> = 8.462×10<sup>-8</sup> sr in a square arcminute. Then multiplying by c gives a speed of 56.75 mph (probably 55 mph, based upon the {{w|National Maximum Speed Law|'traditional' US speed limit}}, before rounding errors in the reverse direction), or 91.33 km/h, showing that you can combine an outrageously high speed with two unnecessary units that cancel each other to form a normal road speed.

It is worth noting that, although some of these examples are ridiculous, there are cases where uncancelled units can be helpful to understanding the concept. For example, while the {{w|Hubble's law|Hubble Parameter}} can be expressed as 2.17132212×10<sup>-18</sup> hertz, expressing it as 67 km/s/Mpc directly relates the quantity to how it is measured and its natural interpretation. Another example would be fuel efficiency in cars, where mi/gal and km/l technically simplify to 1/area, but expressing it in volume and distance allows easy estimations of range and travel cost, while mm<sup>-2</sup> or in<sup>-2</sup> would require significant unit conversions.

==Transcript==
:[White Hat and Cueball are standing to either side of a refrigerator. White hat is lifting on hand up to touch the side of the fridge. The fridge has two compartments, with two doors that open to the top compartment and one bottom compartment, a drawer. The top left compartment has a tall handle on its right, the top right compartment has a tall handle on its left, and the bottom compartment has a long handle on its top. The top left compartment has an oval shape on the top and a paper attached, both with unreadable text. There is also a small square note in the top right corner and an oval shape on the side of the fridge above White Hat. These also have unreadable text.]
:White Hat: This fridge uses only 3 kWh per day!
:Cueball: But will it fit in my kitchen? The ceiling there is only 50 gallons per square foot.

:[Caption below the panel:]
:Pet peeve: Uncanceled units

{{comic discussion}}

[[Category:Pet Peeves]]
[[Category:Comics featuring White Hat]]
[[Category:Comics featuring Cueball]]
[[Category:Math]]