# 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*](https://www.lesswrong.com/posts/xg3hXCYQPJkwHyik2/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 $E=mc^2$, is about $10^{17}$ Joules per kilogram. Compare that to the combustion fuel (H$_2$) with the most specific energy, on the order of $10^8$ J/kg. Nuclear fusion fuel (also hydrogen) can get to $6\times10^{14}$ 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](https://en.wikipedia.org/wiki/Bremermann's_limit), at about $10^{50}$ bits per second per kilogram. Modern computers are much further from this limit than modern energy storage is from $E=mc^2$. Other limits can also be better understood, such as the [Landauer limit](https://en.wikipedia.org/wiki/Landauer's_principle), [Bekenstein bound](https://en.wikipedia.org/wiki/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.[^287iak9l5ea]

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”.[^7428gu3burw] 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.](https://en.wikipedia.org/wiki/Order_of_magnitude#See_also)
    *   [Energy](https://en.wikipedia.org/wiki/Orders_of_magnitude_(energy))
    *   [Power](https://en.wikipedia.org/wiki/Orders_of_magnitude_(power))
    *   [Data](https://en.wikipedia.org/wiki/Orders_of_magnitude_(data))
    *   [Toxicity](https://en.wikipedia.org/wiki/Median_lethal_dose) (median lethal dose/LD50)
    *   [Painkiller potency](https://en.wikipedia.org/wiki/Equianalgesic)
        *   The maximum listed is 1,000,000x as strong as morphine when injected intraspinally
*   [Cell biology by the numbers](https://book.bionumbers.org/)
    *   [Associated website](https://bionumbers.hms.harvard.edu/search.aspx) with citations for ~all the numbers in the book and quite a few outside of it
*   [Neurotech numbers](https://milan.cvitkovic.net/writing/neurotechnology_numbers_worth_knowing/)
*   [Effective Altruism numbers](https://github.com/benthamite/EA-numbers)

[^287iak9l5ea]: 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. 

[^7428gu3burw]: 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.