Magnitudes: Let’s Comprehend the Incomprehensible!
Summary: Comfort with really big and really small quantities is very useful for understanding the world and is perfectly doable with practice. Therefore, you should (1) try to familiarize yourself with the sizes of various things using e.g. spaced repetition and (2) comment some of your favorite numerical facts about the world that give a sense of scale so others can learn from them. I have a list of links at the bottom that I’d love for you to add to, in a similar spirit to The Best Textbooks on Every Subject.
One of the greatest lies that science fiction and popular science told me was that there were features of the universe that are incomprehensibly small or big. Sure, femtoseconds, angstroms, parsecs and electron-volts are foreign to most people— but foreign doesn’t mean incomprehensible. I might not understand Japanese, but that doesn’t mean it can’t be understood. If you talk with physical scientists, you’ll often find that they have a rock-solid grasp on the sort of scales that they deal with, the same way you or I have a solid grasp of how hard it is to move things around on the kilograms/meters/seconds scale we’re so familiar with. I’ve found that a lot of expert intuition takes the form of knowing the sizes and scales of various things. Such knowledge is great for sanity-checking, making connections, visualizing and analogizing. Knowing a single number is trivia, but knowing a web of numbers and their connections to each other is an intuition, or at least a useful part of one.
There are practical advantages to understanding scale, though I’m most interested in refining my intuition. Familiarity with the sizes of things provides a useful starting point for fermi-estimation. It also puts into perspective various (suspected) fundamental limits. For instance, the maximum specific energy, given by , is about Joules per kilogram. Compare that to the combustion fuel (H) with the most specific energy, on the order of J/kg. Nuclear fusion fuel (also hydrogen) can get to J/kg, which is less than 3 OOMs off from the maximum. On the other hand, the maximum possible speed of an information processing device is given by Bremermann’s bound, at about bits per second per kilogram. Modern computers are much further from this limit than modern energy storage is from . Other limits can also be better understood, such as the Landauer limit, Bekenstein bound, and uncertainty principles. Having a general expectation for how big or small things tend to be is also really helpful for spotting anomalies or surprises.[1]
The most efficient way I’ve found to do this is through spaced repetition. I have an anki deck of the sizes of various things. Some questions ask me to provide straightforward answers, for instance “What is the mass of the Earth”.[2] Others ask me to provide analogies, like “if an E. Coli cell is the size of a human, a yeast cell is the size of an elephant and a HeLa cell the size of a blue whale”. I’ve found that learning a bunch of interrelated facts is easier and gives more intuition than learning isolated facts with no real relation to each other. For instance, I also have cards about the radius and density of the Earth, as well as the size, radius and density of other celestial bodies that I can use to put things in perspective.
Now, where can you find these numbers? This isn’t a solved problem, but I’ve found some fantastic sources. Cell Biology by the Numbers is the best book for comprehending the scales relevant in cell biology (and has lots of related resources online). Wikipedia has a series of genuinely fun pages on orders of magnitude. My personal favorites would have to be those on energy, power, information/data, and toxicity of various substances.
Finally and perhaps most importantly, I’d like to solicit anyone reading this post to contribute a few numbers of their own. If you’re uncertain about how representative a particular number is, feel free to put a disclaimer or tell us your epistemic status. Also feel free to put links to various places where you might find these numbers. I’ll keep a list of links below and update them in as comments roll in. I’d love to see what sort of numbers the LessWrong community has on its mind.
Links:
Wikipedia’s mega list of orders of magnitudes of various units.
Toxicity (median lethal dose/LD50)
The maximum listed is 1,000,000x as strong as morphine when injected intraspinally
Associated website with citations for ~all the numbers in the book and quite a few outside of it
- ^
For instance, I had heard many times that fentanyl was really toxic, then kinda just shuffled that qualitative fact away in my brain. Later, when learning about toxicology, I took some time to read over the list of (what feels like) every substance ordered by median lethal dose. While originally I would have guessed that fentanyl was a few times more toxic than some other scary drug like methamphetamine or heroin, it turned out to be multiple orders of magnitude more toxic than either, and more comparable to VX nerve agent or cone snail venom.
- ^
I’d urge you to resist the temptation to memorize numbers to more accuracy than you need. More numbers are often hard to remember and there are rapidly diminishing returns on the intuitions that knowing each digit gives you. If you want precision, just look it up.
The numbers I have in my Anki deck, selected for how likely I am to find practical use of them:
total # hours in a year — 8760
${{c1::200}}k/year = ${{c2::100}}/hour
${{c1::100}}k/year = ${{c2::50}}/hour
# of hours in a working year — 2,000 hours
miles per time zone — ~1,000 miles
california top-to-bottom — 900 miles
US coast-to-coast — 3,000 miles
equator circumference — (before you show the answer, i always find it fun that i can quickly get an approximation by multiplying the # of time zones by the # of miles per time zone!) :::25,000::: miles
US GDP in 2022 — $25 trillion
google’s profit in 2022 — $60 billion
total US political spending per election — ~$5 billion
median US salary in 2022 — $75k
LMIC’s GDP per capita in 2022 — $2.5k
world population in 2022 — 8 billion
NYC population in 2022 — 8 million
US population in 2022 — 330 million
The energy stored within the nitrogen triple bond, one of the strongest common bonds in chemistry, is ~10 eV which is a bit more than 15*10^-19 J. This is considered *very stable*. It is quite the feat for some biological processes to be able to break this bond. Now, an average human punch confers around 150 J of energy. So, if you had some very strange means of directing that energy, punching the air could split around 10^20 nitrogen molecules which is around 4 ml (4.65 mg) at room temperature.
I have an Anki deck in which I’ve half-heartedly accumulated important quantities. Here are mine! (I keep them all as log10(value in kilogram/meter/second/dollar/whatever seems natural), to make multiplication easy.)
We can even produce small amounts of anti-hydrogen, but not as a fuel.
While good to have for reference, I didn’t find the enormous tables on Wikipedia helpful. Using the mass of the Earth as an example, I have no intuition for the number 1024 : it’s just ten apples with twenty-four smaller apples floating next to them (relevant xkcd). And I don’t think memorizing these numbers, with spaced repetition or otherwise, is all that helpful for intuition-building.
For astronomical distances, what I have found helpful is to do everything in terms of the speed of light (108.5 m/s = a foot per nanosecond). The sun is 8 minutes = 500 seconds away. The moon is 1.5 seconds away. Jupiter is 45 minutes away, and doubles every step to Saturn, Uranus, and Neptune.
For the circumference of the Earth, the distance from the pole to the equator is a nice, round, 10,000 km. (Remarkably convenient coincidence!)
For masses, use the density of water: the Sun and all the gas giants are the same density as water. Earth is the density of rock, about √10 that of water. The density of metal is 10 times water.