3038: Uncanceled Units

2025-01-15T17:16:04Z

Tomorand: /* Explanation */

{{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 PLANCK CONSTANT, WHICH IS TECHNICALLY A FREQUENCY AND CAN THUS BE EXPRESSED IN HERTZ - 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.

[[White Hat]] is presenting a refrigerator to [[Cueball]], claiming it only uses 3 kWh per day. Kilowatt-hours is a commonly used unit of electrical energy in the United States, being the amount of energy consumed by one kilowatt of power usage for one hour. As the unit in which power rates are typically reported and bills calculated, it's the most useful piece of information to the average consumer. This measure, however, could be simplified, since a kilowatt is equivalent to a kilojoule per second. Rather than adding a second unit, energy could simply be reported in megajoules (1 kilowatt-hour is 3.6 megajoules, so the refrigerator uses 10.8 mJ per day).

Appliances, however, frequently report their typical power use in terms of kilowatt-hours per day (or per month or year). Once again, this is useful to consumers, because it makes it easier for them to understand how much money it will cost them to run. However, it's an inelegant way to use units, because it uses power, which is already a measure of energy per time, multiplies it by one time unit, and then divides it by another. This clunky method apparently chafes at [[Randall]]'s mind, possibly due to his scientific background (which encourages simplifications of units). The "hour" and "day" terms, both being units of time, can simply cancel out by dividing the whole number by 24, meaning that the refrigerator averages 0.125 KW, or 125 watts, to run. It should be noted that this doesn't mean the refrigerator constantly draws this amount of power, since the compressor in the refrigerator only runs intermittently, but running it over the course of a day (with typical use) is expected to give that average power use.

Cueball (possibly 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 weird collection of uncancelled units. Gallons can be transformed to cubic feet (1 US gal ≈ 0.1337 ft3), which can be divided by the square feet, yielding a ceiling height of around 203.7 cm, or around 6 feet 8 inches. (Using imperial gallons [1 UK gal ≈ 0.1605 ft3], the height is approximately 244.7 cm, roughly 8 feet.) This is intended to lampoon the use of uncancelled units by showing how odd things become if they're generally used.

''[[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).

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 customary units which support even divisibility 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 arcminutes2 per steradian, where c is presumably the speed of light in vacuum, 2.998×108 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))2 = 8.462×10-8 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, uncancelled units can be helpful to better understand the concept, the {{w|Hubble's law|Hubble Parameter}} can be expressed as 2.17132212×10-18 hertz, 67 km/s/Mpc is directly related to how it is measured and gives a better understanding of what it means. Another example would be fuel efficiency in cars, as mi/gal and km/l technically simplify to area, but by expressing it in volume and distance it allows easy estimations of range and travel cost, while mm2 or in2 would require significant unit conversions.{{cn}}

==Transcript==
{{incomplete transcript|Do NOT delete this tag too soon.}}

:[White Hat and Cueball are standing to either side of a refrigerator. The fridge has two top compartments and one bottom compartment. 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 a paper attached to it with unreadable text, possibly an advertisement.]
: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:Comics featuring White Hat]]
[[Category:Comics featuring Cueball]]
[[Category:Math]]
[[Category:Pet Peeves]]

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3038: Uncanceled Units