In physics we are often faced with unbelievably large numbers. In fact, studying systems with a huge number of objects is a sub-genre in itself called statistical physics. Notably, some numbers become so large that we can’t really tell them apart [1]. There is a point when addition just doesn’t make any difference to a large number. Can you tell the difference between 10^{23} = 100 000 000 000 000 000 000 000 and 10^{23} +1 = 100 000 000 000 000 000 000 001? For all intents and purposes, they are the same, right? In fact, there is even a point when multiplication doesn’t change the size of really big numbers. Is it possible to tell the difference between 10 x (_{10}10^{23}) and _{10}10^{23}? _{10}10^{23} is a 1 followed by 100 000 000 000 000 000 000 000 zeros, and 10 x (_{10}10^{23}) is the same with another zero tagged on the end. The two numbers indisputably differ by a factor 10, but they are both so big that it hasn’t really changed much about what we understand about their size.

This rather flippant attitude towards numbers has actually been well documented. It’s called ‘logarithmic number space mapping’ [2]. This means that our APPRECIATION of a number’s size doesn’t vary in the same way as the ACTUAL value of the number. More specifically, we only increment the value of our appreciation when the number’s actual value becomes an order of magnitude larger, that is to say when we multiply it by 10.

This trick is really useful from an evolutionary perspective, because it’s a very effective way of making decisions based on a quick estimation of how many things there are: “There is 1 worm in my fruit.”, “I need 2 friends to help me climb this rock.”, “Which of these 5 wildebeest looks juiciest to hunt?”, “Uhoh, there are lots of wildebeest about to trample me!”. It’s not really useful to know how many wildebeest there are, just an idea of how many you can handle. Fortunately for us, we no longer need to deal with wildebeest migrations. We do however need to make decisions based on the ‘linear number space mapping’ (the number line with nice evenly spaced numbers) we learnt at school. We need to be precise and quantitative: if I have 500€ in my bank account, I can’t afford a 600€ computer even if 500 and 600 are basically the same in terms of order of magnitude. In this world our logarithmic demons come to haunt us. Here are two examples which I’ve encountered this week.

The first is an e-mail I received from the French train service SNCF, claiming that by using the train you emitted 32 times less CO2 equivalent by kilometer than by using the car [3]. No new ideas there, except that I realized that figuring out what 32 times something actually means isn’t so straight forward. How big is 32? Well quite small when you compare it to 1000 (about 32 times smaller in fact!), but quite large compared to 1 really. I had a good feeling for the order of magnitude, but I couldn’t visualize it concretely. So, I started comparing. Let’s say you have an emission budget which is fixed, how far you can travel using the train or car? Typically, if you commute to work, you might live in suburbs and have to drive 20km to work each day (40km there and back). How far can you travel by train for the same carbon budget? 20*32 = 640 km. That’s the equivalent of Lyon – Barcelona! Now I had created a quantitative appreciation for 32, not just a feeling.

If I had trouble imagining the size of even ‘small’ numbers like 32, what about 1 billion for example? Jeff Bezos is estimated to have a fortune of 200 billion dollars approximately. How much money is that exactly? Let’s say that for every one of Jeff Bezos’ dollars you walked 1 meter (one big step) every second. How far would you be able to go? How long would it take you? If you walked all the way to the sun (150 million kilometers), which I think we can all agree is already almost unfathomably far away, you would still have a quarter of your money left to spare. Oh, and if you arrived there now, you would have started when the Pyramids were being built. Another way of looking at it, is that you would need to find a way to spend 1 million dollars every 4 hours for 80 years to spend 200 billion dollars. It’s a lot of money.

Aside from my personal opinion on the examples above, I think very little time is taken to train us to be aware of the size of numbers and what they mean. You can have perfectly valid reasons to choose to take the car 20km instead of wanting to travel 640km by train, I don’t think there is necessarily a wrong answer there. But if you don’t understand quantitatively what you are sacrificing (for yourself, for others) by choosing one or the other, it seems impossible to make an informed decision. My tip: when faced with any number, ask yourself “Can I visualize this number? Do I know what this means?” If not, try to break it down into something you can imagine (distance, time, the number of times you had to wash your teeth last week…) and see how you react!

[1] In fact I would argue that Nature can’t tell them apart either. Leave a comment if you want to discuss.

[2] S. Dehaene et al., “Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures”, Science (2008)

[3] This of course depends on the numbers you are using. I checked the source of the SNCF’s math (https://www.bilans-ges.ademe.fr/fr/accueil) and the comparison they make is in their favor but not discreditably so.

To go further:

– An excellent (if partisan) visualisation of wealth concentration in the United States https://mkorostoff.github.io/1-pixel-wealth/

– A much deeper look into the fascinating concept of large numbers by Scott Aaronson https://www.scottaaronson.com/writings/bignumbers.html

A very thoughtful read, thanks. Regards visualizing numbers to get a ‘feel’ for them, our perception also depends on context. Let’s use current Covid numbers to illustrate: The death rate world-wide for recorded infections is a little over 2.5% or around 1 in 40. Now I’d be willing to buy a lottery ticket with only a 1 in 40 chance of winning. But if I knew that, on average, every 40 times I crossed the road I would be killed by a passing vehicle, I would reconsider why I wanted to cross !

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Definitely!

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Great article, and good call to give some everyday life points of comparison to big numbers. For the French speaking readers, I recommend the video series by Science Etonnante on Youtube (https://www.youtube.com/watch?v=MCg2lw4Nxno&ab_channel=ScienceEtonnante) called “crétin de cerveau”, where they explain why humans are so bad at estimating small and big numbers, especially when dealing with probabilities (see Peter’s example) and how it can be used to manipulating people.

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C’est vrai qu’elle est géniale cette vidéo. Le truc incroyable avec cette idée d’utilité perçue et pas objective, c’est qu’elle n’est pas symmétrique de parts et d’autres du gain 0. En effet, à perte et gain objectif identique en valeur absolue (c’est-à-dire -5 euros de perte ou +5 euros de gain) la perte est perçu comme apportant plus de dissatisfaction que le gain n’apporte de satisfaction. Ça à des implications dans plein de domaines, dont le tourisme. Lors de mes formations pour être moniteur de ski ils nous ont dit qu’il faut 12 expériences positives pour compenser une seule expérience négative!

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