Brainteaser for anyone who doesn't regularly think about units.
Why is it that I can multiply or divide two quantities with different units, but addition or subtraction is generally not allowed?
According to AnthonyC, units, for some reason, behave like variables in algebra, e.g.
- simplifies to , with arithmetic this computes to .
- simplifies to , with arithmetic this computes to
- or don't simplify so that we could perform addition or subtraction.
So this algebraic behavior does indeed look like what we would expect from units.
For those who would like a hint.
In English, "And" generally indicates addition, "Per" division.
Now consider which of the following makes sense:
Ferrets and seconds
Ferrets per second
1I feel like this is mostly an artifact of notation. The thing that is not allowed with addition or subtraction is simplifying to a single term; otherwise it is fine. Consider:
10x + 5y -5x -10y = 10x - 5x + 5y -10y = 5x - 5y
So, everyone reasons to themselves, what we have here is two numbers. But hark, with just a little more information, we can see more clearly we are looking at a two-dimensional number:
5x - 5y = 5
5x = 5y +5
5x - 5 = 5y
x - 1 = y
y = x - 1
Such as a line.
This is what is happening with vectors, and complex numbers, quarternions, etc.
I think the way arithmetic is being used here is closer in meaning to "dimensional analysis".
"Type checking" through the use of units is applicable to an extremely broad class of calculations beyond Fermi Estimates.
Stephen puts it elegantly. Though for me who is more of a code monkey, I'd like to think of it as "Runtime Non-Zero cost type safety through some const generics".
So, agreed with this all being an excellent world modeling technology and it would be great if more humans felt comfortable using it. And I agree that most humans are not comfortable using arithmetic.
However, I don't think units are part of arithmetic. I think they're part of algebra. They're like variables that you know never equal zero.
Now, if you know arithmetic, there's really only a couple of things algebra tells you you're allowed to do, and most people probably never internalize them: you can always add zero or multiply by one for any quantity, and you can multiply by or add a constant to both sides of an equation. That's basically it. Using units is all about creative ways to multiply by one.
Also: A significant part of my professional life gets called "modeling" but is really just things like this, in the form of building spreadsheets of constants and assumptions to make rough estimates of quantities we can't measure, then sanity checking them. It's the kind of thing even many professional scientists and engineers and investors don't feel comfortable doing outside their own specialties.
2Edit to add: in college my first physics class assignment was just random Fermi problems to be answered without looking anything up using top-of-mind round-number assumptions (1, 2, or 5 x 10^n). Things like "How many feathers does a chicken have?" or "How many letters are there in every book in the main campus library?" Second favorite homework assignment I've ever been given.
Most people never realize how much the things they know imply about the things they don't.
May I throw geometry's hat into the ring? If you consider things like complex numbers and quarternions, or even vectors, what we have are two-or-more dimensional numbers.
I propose that units are a generalization of dimension beyond spatial dimensions, and therefore geometry is their progenitor.
It's a mathematical Maury Povich situation.
I am honor bound to mention that we do use gravity to store energy - https://en.wikipedia.org/wiki/Pumped-storage_hydroelectricity Big fan of the blog.
We also literally use large weights. The most promising proposals use existing infrastructure like train tracks and mineshafts, eg Gravity System Aids Storage in Unused Mine Shaft - News.
Interesting. Looks like they are starting with a deep tunnel (530 m) and may eventually move to the deepest tunnel in Europe (1444 m). I wish I could find numbers on how much weight will be moved or the total energy storage of the system. (They say quote 2 MW, but that's power, not energy—how many MWh?)
According to this article, a Swiss company is building giant gravity storage buildings in China and out of 9 total buildings, there should be a total storage of 3700 MWh, which seems quite good! Would love to know more about the technology.
It happens to be the case that 1 kWh = 3,600,000 kg/s². You could substitute this and cancel units to get the same answer.
This should be kg m²/s².[1] On the other hand, this is a nice demonstration of how useful computers are: it really is too easy to accidentally drop a term when converting units
- ^
I double checked
You're 100% right. (I actually already fixed this due to someone emailing me, but not sure about the exact timing.) Definitely agree that there's something amusing about the fact that I screwed up my manual manipulation of units while in the process of trying to give an example of how easy it is to screw up manual manipulations of units...
You mentioned a density of steel of 7.85 g/cm^3 but used a value of 2.7 g/cm^3 in the calculations.
BTW this reminds me of:
https://www.energyvault.com/products/g-vault-gravity-energy-storage
I was aware of them quite a long time ago (the original form was concrete blocks lifted to form a tower by cranes) but was skeptical since it seemed obviously inferior to using water capital cost wise and any efficiency gains were likely not worth it. Reading their current site:
The G-VAULT™ platform utilizes a mechanical process of lifting and lowering composite blocks or water to store and dispatch electrical energy.
(my italics). Looks to me like a slow adaptation to the reality that water is better.
You mentioned a density of steel of 7.85 g/cm^3 but used a value of 2.7 g/cm^3 in the calculations.
Yes! You're right! I've corrected this, though I still need to update the drawing of the house. Thank you!
Is the main point of this post that people should play around more with numbers and estimation? If so, then I agree with it, but there are two aspects of the post that I found distracting.
One was the overly broad use of the word "arithmetic". Arithmetic and algebra have substantially different histories, and they occupy somewhat different roles in contemporary society. (Especially for young math students.) Consequently, I think it's best to avoid using the words interchangeably.
The other is the repeated emphasis of dimensional analysis. Although it was probably worth mentioning once, it doesn't appear to be any more relevant than methods for sanity-checking literal arithmetic, and I don't think that either is central to your epistemological claims.
Calculations on Hydroelectric Energy Storage
For those interested in the numbers on pumped hydroelectric storage, we can get more energy by increasing 'head' or the distance that the weight falls, from 6 meters to up to 500 meters for some of the largest projects (and we could in theory go bigger).
Let's pick a more reasonable number like 60 meters:
MASS/house = 15 kWh/house / (9.8 m/s² × 60 m) = 91,836 kg/house = 91 m^3/house
Let's say we have a dam with ~20 meters of water level fluctuation (drawdown). Then that's 5 m^2 per house of surface area.
As a sanity check, Bath County Pumped Storage Station in VA stores about 24000 MWh/ 30 KWh/house = 800,000 houses worth of energy.
800,000 houses * 5 m^2 = 4 km^2
The Bath County reservoir is about 1km^2 so we're in the right range here (the reservoir has a little more drawdown and a way bigger head).
Curated. This post hit me exactly at a moment where I was trying to become a more "numerate person" (i.e. gain some skills where it was easier to convert problems into 'things with numbers' and then solve the problems with simple math).
I liked the worked examples of the post, and I appreciated the emphasis on why it's important to keep track of units (which I had roughly known, but seeing how it played out in the examples drove the point home more).
Incidentally, female chimps seem to live 25% longer than males—imagine human women lived until 90 while men died at 71.
Arithmetic error? both 71*1.25 and 71.5*1.25 to the nearest integer are 89, not 90. The error might (low-confidence, 10%) have been caused by calculating a 12.5% increase of 80 (exactly 90) and also dividing 80 by 1.125 (~71.1).
Well done, yes, I did exactly what you suggested! I figured that an average human lifespan was "around 80 years" and then multiplied and divided by 1.125 to get 80×1.125=90 and 80/1.125=71.111.
(And of course, you're also right that this isn't quite right since (1.125 - 1/1.125) / (1/1.125) = (1.125)²-1 = .2656 ≠ .25. This approximation works better for smaller percentages...)
I think that I have a personal example of this advice in action. I often find it helpful to use my driving speed and the distance to the next turn to estimate how soon I'll need to turn. That indicates how desperately I need to change lanes, whether it's a good time to initiate a conversation, etc.
Society seems to think pretty highly of arithmetic. It’s one of the first things we learn as children. So I think it’s weird that only a tiny percentage of people seem to know how to actually use arithmetic. Or maybe even understand what arithmetic is for.
I was a bit thrown off by the seeming mismatch between the title ("underrated") and this introduction ("rated highly, but not used or understood as well as dynomight prefers").
The explanation seems straightforward: arithmetic at the fluency you display in the post is not easy, even with training. If you only spend time with STEM-y folks you might not notice, because they're a very numerate bunch. I'd guess I'm about average w.r.t. STEM-y folks and worse than you are, but I do quite a bit of spreadsheet-modeling for work, and I have plenty of bright hardworking colleagues who can't quite do the same at my level even though they want to, which suggests not underratedness but difficulty.
(To be clear I enjoy the post, and am a fan of your blog. :) )
Of all the cognitive tools our ancestors left us, what’s best? Society seems to think pretty highly of arithmetic. It’s one of the first things we learn as children. So I think it’s weird that only a tiny percentage of people seem to know how to actually use arithmetic. Or maybe even understand what arithmetic is for. Why?
I think the problem is the idea that arithmetic is about “calculating”. No! Arithmetic is a world-modeling technology. Arguably, it’s the best world-modeling technology: It’s simple, it’s intuitive, and it applies to everything. It allows you to trespass into scientific domains where you don’t belong. It even has an amazing error-catching mechanism built in.
One hundred years ago, maybe it was important to learn long division. But the point of long division was to enable you to do world-modeling. Computers don’t make arithmetic obsolete. If anything, they do the opposite. Without arithmetic, you simply can’t access a huge fraction of the most important facts about the world.
The magic lives in a thing called “units”.
Chimps
It’s amazing how much we don’t know about nutrition. For example, would you live longer if you ate less salt? How much longer? We can guess, but we don’t really know.
To really be sure, we’d need to take two groups of people, get them to eat different amounts of salt, and then see how long they live. This is expensive, ethically fraught, and runs into the problem that when you tell people to eat differently, they usually ignore you.
So I’ve often wondered: Why don’t we do these experiments on animals? Why not get two big groups of chimpanzees, and feed them different amounts of salt? Chimps aren’t people, but it would tell us something, right?
Why don’t we do this? Because arithmetic.
How much would such a study cost? To figure this out, you will need three numbers:
Let’s do these. First, how long do chimps live? In captivity the average seems to be around 36.3 years. (Incidentally, female chimps seem to live 25% longer than males—imagine human women lived until 90 while men died at 71.)
Second, how much does it cost to maintain a chimp? Capaldo et al. looked at the average costs in various research facilities in the US in 2009. They estimate around $75/day (in 2024 dollars).
Finally, how many chimps do you need? To calculate this, you should do a “power calculation”—you guess how much life expectancy varies due to (a) salt and (b) all random factors, and work backwards to see how many chimps you need to separate the signal from the noise. There are lots of calculators for this. If you assume chimps live 36.3±8 years and salt would change life expectancy by 2 years, these will tell you that you need 502 chimps.
So now we can do our calculation:
502 chimps
× 36.3 years
× 365.25 days / year
× 75 dollars / (chimp day)
≈ 499,185,349 dollars
Notice three things.
First, 500 million dollars is a lot. That’s five times what the big alcohol trial would have cost. It’s a gigantic amount of money for something that would only give indirect evidence for the impact of salt in humans, and wouldn’t even do that until decades in the future.
Second, notice how I kept the units. Always keep units! On the “top” of the calculation, the units were “chimps × years × days × dollars”. On the “bottom”, the units were “years × chimps × days”. When you cancel terms, you’re left with dollars only. Units are great because if you made a mistake, it will probably show up in the units not working out. We’ll see other benefits below. So: ALWAYS KEEP UNITS.
(If you think you’re an exception and you don’t need units, then you especially need to keep units.)
Finally, notice that this calculation didn’t just tell us how expensive the study would be. It also points towards why it’s so expensive, and what would be needed to make it cheaper.
One option would be to try to get away with fewer chimps. The reason so many are needed is because the likely impact of salt is pretty small compared to natural variation in life expectancy. You might be able to reduce that natural variation by, for example, using pairs of chimp twins to eliminate genetic variation. If that reduced the standard deviation from 8 years to 5 years, then you’d only need 196 chimps and the total cost would be “only” 195 million dollars. Sounds nice, though I imagine that creating 98 chimp twins wouldn’t be free.
Another option would be to reduce the cost of maintaining chimps. Doesn’t $75 per chimp per day seem very expensive? Perhaps you could find a way to use existing chimps in zoos? Or you could use dogs instead of chimps and offer dog owners subsidized dog chow with slightly varying salt levels? Or you could built a gigantic outdoor facility with 50,000 chimps where you could amortize costs by running 100 experiments in parallel?
I’m not sure which (if any) of these options would work. My point is that doing the arithmetic quickly takes you into specifics about what would be necessary to actually move the needle. Without doing the arithmetic, what chance would you have to say anything meaningful?
Big blocks
If I know my readers then at some point in your life you probably considered using gravity to store energy. Maybe you can put solar panels on your roof, but instead of storing their energy in batteries, you can just lift up a giant block into the air. At night you can slowly let the block down to power your house. How big a block do you need?
Let’s assume you don’t know much physics.
To answer this question, you’ll need two numbers:
If you check the internet, you’ll learn that the average US household uses around 30 kWh of energy per day. Now, what’s a “kWh”? To you, person who doesn’t know much physics, it looks scary, but apparently it’s some kind of unit of energy, so let’s just write it down. Assume you need to store half your daily energy for usage at night, or 15 kWh.
Now, how much energy can you store by lifting a giant block up into the air? A little bit of searching reveals that “potential energy” is the product of mass, gravity, and height: If you lift a block of weight MASS up to height HEIGHT, the stored energy is U=MASS × g × HEIGHT where g ≈ 9.8 m/s² on Earth. Your house is 6m tall, and you reckon that’s as high as you could lift a block, so you use h = 6m. Thus, the amount of energy you can store is
MASS × (9.8 m/s²) × 6 m.
What now? Now, you’re done! You just equate the energy you need to store with the energy you can store with a block that weighs MASS:
15 kWh = MASS × (9.8 m/s²) × 6 m.
Is this frightening? There are units everywhere. You never figured out what a kWh is. How is that related to meters and seconds? What does it mean to square a second? Panic!
Relax. We have computers. You can just mechanically solve the above the above equation to get MASS = 15 kWh / (9.8 m/s² × 6 m) and then literally type that into a search engine to find that MASS is:
Look at that—the answer is in kilograms!
It happens to be the case that 1 kWh = 3,600,000 kg m²/s². You could substitute this and cancel units to get the same answer. But don’t. Attempting that just gives you the chance to screw things up. Why complicate your life?
And as before, the units give you a sort of “type checking”. If your calculation was wrong, you’d have to be very unlucky to get an answer that was in kg anyway.
Here the units did most of the work for you. So it’s a good thing you kept units.
ALWAYS KEEP UNITS.
More big blocks
So, a 918 thousand kg block. How much would that cost? It seems natural to use rock, but it’s hard to get million kilogram boulders delivered to your house these days. So let’s use steel. Current steel prices are $350/ton. So we want to solve
918,367 kg = MONEY × 1 ton / $350.
How are tons related to kilograms? Say it with me: Not your problem. Just solve the above equation for MONEY and ask the big computer to learn that MONEY is
That’s 65× more than just buying a 20 kWh home battery. But let’s say you’re committed to the bit. How big would that block be? Some searching reveals that the density of steel is around 7.85 g/cm³. So if you have a cubic block of volume VOLUME, then
MASS = 7.85 g / cm³ × VOLUME.
Solving for VOLUME, using the previous value for MASS, and not stressing about units, you can easily find that VOLUME is
A 117 cubic meter block is around 4.9 meters on all sides. So, roughly speaking, your house will look something like this:
As it happens, 1 million kg cranes do exist. But even used, they’ll set you back another million dollars or so. If you’re going to get one of those, then may I suggest that the same weight is given by almost exactly 4.5 Statues of Liberty? So I suggest you also consider this option (drawn to scale):
Either way, your neighbors will love it.