Uncanceled Units
Speed limit c arcminutes^2 per steradian
Speed limit c arcminutes^2 per steradian
Another of Randall's 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 watts (a unit for the rate of energy transfer, equal to 1 joule per second).
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, 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. Also, "per day" makes it clear that this is the average power usage rather than the maximum power usage (they are different because refrigerators cycle on and off throughout the 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.
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 in3), 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 in3], 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 in3/gal * (1/12 ft/in)3 / 1 ft2 = 6.68 ft = 6 ft 8 in
This can be understood as "the height such that every square foot of ceiling has 50 gallons under it." The what if? article 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). In 2327: Oily House Index, volume is also divided by area to get house height.
A common source of unit drama occurs between lay people who are looking for everyday 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 arcminutes2 per steradian, where c is presumably the speed of light in a 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 '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 Hubble Parameter can be expressed as 2.17132212 × 10-18 Hz, 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-2 or in-2 would require significant unit conversions. Another example might be the units of the gravitational constant G, ~6.674 × 10-11 m3/(kg×s2), which might be written as (m/s2)/(kg/m2), although this only involves unjumbling rather than uncancelling units.