Review

Humans can’t grasp large numbers. True, when we hear “one hundred”, we might imagine ten rows of ten or a few written paragraphs. Some of the more number-savvy might hear one thousand and see half the stars in the sky. But when we reach for higher powers of ten—ten thousand, one hundred thousand, and the formidable -illions—we come up short. These numbers are beyond the reach of our intuition. Our innumeracy leads us to undervalue big issues because we can’t visualize just how big they are (scope insensitivity), making utilitarian calculations difficult.

However, humans make up for a lack of inborn numeracy with a talent for creating and manipulating mental images. This opens a shortcut where we use our natural visual capabilities to improve intuition with numbers.

Large Numbers

One remedy for number numbness is chunking. Chunking happens when we mentally represent a quantity by comparing it to a reference that’s easy to visualize, as in, "one megaton of TNT could destroy Paris, so twenty well-placed megatons will destroy a country." Writers litter comparisons like this in news and popular science articles because they help readers grasp the wide-ranging quantities of science and mathematics.

These comparisons are helpful when writers supply them, but they're no solution we can broadly apply. A single repository of these comparisons that we can memorize would help in more situations.

This is an attempt at such a repository.

Instead of trawling the internet for poorly sourced examples, we can multiply tiny units of volume by powers of ten and visualize the resulting volume. I tried two different units for this: raindrops and grains of sand. These units are useful because they never change, unlike population, and because we can easily visualize sand grains and raindrops, unlike seconds, meters, dollars, years, or breaths.

For each power of ten between one hundred and one trillion, the chart below gives the volume filled by that many grains of sand and that many raindrops. I’ve included links to images when possible to help visualization. For smaller numbers, I’ve also thrown in a few miscellaneous examples, though these become hard to find as the numbers grow.

Power of TenShort Scale NameVolume Filled by That Many Grains of SandVolume Filled by That Many RaindropsMiscellaneous
One HundredA tiny wispTwo-thirds of a US teaspoon (3.2 mL)
One ThousandA tiny pilePerfume bottle
  • Words on two standard single-spaced pages
  • Half the number of stars visible to the naked eye in ideal conditions
  • Years between the first use of gunpowder in warfare and the Napoleonic Wars
Ten ThousandA small pileClassic coca-cola bottle
  • Diameter of the circle of the horizon at sea level in meters
  • Age of the city of Jericho, the oldest city known
One Hundred ThousandOne-twentieth of a marshmallow vLarge water bottle
  • Entries in all 2,340 pages of the Collins English Dictionary
  • Breaths a human takes in four days
One MillionHalf a marshmallowMotorcycle caseCharacters in a 335-page book
Ten MillionA cola mini Tall refrigerator
 
One Hundred MillionTea kettleSmall Jacuzzi
 
One BillionEight cola bottlesWater tankHighest number countable in a lifetime
Ten BillionOne fifth of a hot tubOne-tenth of an Olympic swimming pool 
One Hundred BillionTwo hot tubsEnough to fill the interior of one floor of 432 Park Avenue
 
One TrillionPetroleum TankerNine floors of 432 Park Avenue
 

 

I’m open to suggestions for new entries.

Fractions

If we can get big, why not also get precise? The difference between 36% and 40% is important, but hard to visualize. Representing probabilities as ratios helps: 40% is two in five, while 36% is nine in twenty-five.

But how much is nine in twenty-five? Is that more or less than seven sixteenths? Fractions with high denominators are barely more legible than decimals. If we don’t require exact precision, we can round the ratio to a fraction with a lower denominator. Ten seems a reasonable number.

For all percentage values to two significant figures, this chart gives an estimate as a fraction with a denominator no greater than ten. Print it out and stick it on your wall. The next time you see 36% in the wild, glance at the chart and say, “oh, that’s about three eights,” and thus be enlightened. With time and practice, you’ll be able to do this without the chart.

The small-denominator restriction does introduce limitations. Simple fractions aren’t great for values less than 8% or greater than 92%, but they are good enough for values in between.

If denominators up to ten are too imprecise (or too precise) for you, here's a program to generate the nearest fraction of any given maximum denominator to any given decimal.

Methods 

Sand Scale

I model a grain of sand as a sphere with a radius of 0.18 millimeters, which is typical. This makes the volume of a single grain 0.0244 cubic millimeters. However, it’s not as simple as multiplying that number by the number of grains, because spheres don’t pack perfectly. In real life, sand has porosity: I use around 30% porosity (34%), but it can range from 25.6% to 43.4%.

Droplet Scale

A raindrop is modeled as a sphere with radius 1.989 mm for a volume of 33 cubic millimeters. Note that a raindrop is about two orders of magnitude larger than a grain of sand.

Sources can be found here.

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3 comments, sorted by Click to highlight new comments since: Today at 8:46 PM

I think physical size of things isn't the best mnemonic for numbers, because we get confused by the difference between length, area, and volume/weight. For example, a cruise ship is 60 times longer than a minivan (5m vs 300m) but 100000 times heavier (2T vs 200KT). So which number should jump to mind when imagining their relative sizes, 60 or 100000 or something in between?

My favourite way to imagine large numbers is to use units of time instead. A thousand seconds is about 15 minutes, a million seconds is about 10 days, and a billion seconds is about 30 years. It's really easy to remember.

It's especially vivid when thinking about population or death toll numbers. If 1 person = 1 second, then 9/11 was about an hour, the Afghan war was two days, the Vietnam war was a month, and WWII was two years. Covid is two months so far, AIDS was a year, the Spanish flu was two years, the Black Death was three years. NYC population is three months, US population is ten years, world population is two centuries and a half, all people who ever lived are three thousand years.

I heard an advice (sounds reasonable, but I didn't try it) that if you want to educate people about national economy without them getting confused by the "-illions", you should pick one unit, for example "millions of dollars", and always express costs using this unit. Importantly, do not call it "millions of dollars" but "megadollars", to make it one word, and thus prevent people from converting it back mentally.

For example, instead of "$1,000" always say "0.001 M$"; instead of "$1,000,000,000" always say "1000 M$". The idea is that after using this style consistently, people will remember the difference between prices expressed in "fractions of megadollars" and prices expressed in "thousands of megadollars" (where they would otherwise confuse e.g. millions with billions), because they always heard it that way.

It would work best if this style was adopted by schools and journalists consistently. (Yeah, one can dream.)

Big thanks for doing this for large magnitudes!

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