We come from the future
We come from the future

​How to Comprehend Incomprehensibly Large Numbers

We humans are a smart bunch, but we really suck when it comes to understanding and handling excessively large numbers. Here’s why we’re so bad at it — and what you can do to make sense of concepts and figures that are unreasonably huge.

To learn more about grasping and processing large numbers, I spoke to mathematician Spencer Greenberg, co-founder of the A.I. powered hedge fund Rebellion Research, and founder of ClearerThinking.org, an online project offering free, interactive training programs that help people enhance their decision making skills.

A Cognitive Limitation

It shouldn’t be too surprising that humans have great difficulty with large numbers. While living and evolving in a so-called state-of-nature, our paleolithic ancestors had no need (i.e. no environmental pressures) to develop such a capacity. Back then, and prior to the advent of a formal numbering system, early humans only really needed to get a basic sense of small batches of quantities, like the number of people in the clan, or how many animals might occupy a certain area.

These days, however, we’re surrounded by large numbers. Like, stupid large numbers. We’re told that there are 7-billion humans on Earth, that there are 300-billion stars in the Milky Way, and that there may be upwards of 70-sextillion stars in the Universe (that’s 1021, or a 1 with 21 zeros behind it). Good luck trying to wrap your head around what such a quantity actually means or signifies.

In addition to this cognitive limitation, and as Greenberg pointed out to me, some languages don’t even distinguish large numbers from each other at all. Take the Pirahã language, for example, a dialect of an indigenous hunter-gatherer tribe from the Amazon. This language is devoid of words for precise numbers, but instead has concepts for ‘a small amount’ and ‘a larger amount.’

“So it seems humans can survive well — in at least some, maybe most or all — natural environments without differentiating large from very large,” Greenberg told io9. “This bolsters the argument that it really is not essential from an evolutionary perspective.”

But that doesn’t mask the fact that humans are absolutely awful with large numbers. Once numbers get beyond a certain point, they tend to lose all meaning.

“We can easily visualize five things,” says Greenberg. “We can even roughly visualize approximately 100 things — by, say, picturing a large crowd gathered. But when we’re talking about millions of things our ability to visualize completely fails.” He says that trying to imagine a million people is about as useless as trying to imagine a hundred million.

He adds that, by using numbers for our whole lives, we’ve built up our intuitions to “feel” that 10 is a lot bigger than one, and that 100 is a lot bigger than 10. But we have no such “feeling” about a quadrillion versus a trillion. In fact, we might not even know which is bigger. Even when we see them written out:

1,000,000,000,000,000 versus 1,000,000,000,000

Indeed, our eyes could easily miss that one is a thousand times bigger if we’re not paying close attention. But thankfully, there are some tricks we can use to help us make better sense of numbers like these.

Break It Down

Greenberg says it’s a good idea to convert large numbers to a different unit to make the large numbers smaller.

For instance, the U.S. national total public debt is about \$17 trillion. That number is nearly impossible to understand intuitively if left in units of dollars. But there are two much easier ways to think about it.

“With about 317 million people in the U.S., that comes out to \$54 thousand per person,” he says. “That’s a number that’s much easier to make sense of — there is \$54 thousand dollars debt owed for each person.”

The other way to think more intuitively about this is that the U.S. has about \$16.8 trillion in GDP, so the debt is about equal to the total market value of all final goods produced in the US in a single year.

But Greenberg says we should avoid units that don’t have any relevance, like “A billion pennies on top of each other forms a tower that’s about 870 miles high.”

Change the Unit of Measurement

We’re often confronted with excessively large figures that would be better expressed with a different unit, like switching from feet to miles (or meters to kilometers), from ounces to pounds, from seconds to years, and so on.

“For instance, if someone says that the Mariana Trench reaches to a maximum depth of 36,000 feet, that’s tough to make sense of,” says Greenberg. “It’s much easier for our brains to understand that as 6.8 miles (11 km), which is a distance we already have a pretty good intuition for.”

That said, be sure to avoid units for which you don’t have an intuitive grasp. Most people, for example, have little intuitive sense of how much a tonne of weight is.

Convert Large Numbers to Batches You’re Familiar With

What does it mean to think about 400,000 people? One thing I like to do is break it down into something I’m familiar with and that I can kind of visualize: the crowd attending a sporting event. For example, hockey arenas seat about 20,000 people. So, you could envision 400,000 people as 20 hockey arenas worth of people.

Tumar/Shutterstock

Or as Greenberg told me, if you have \$100,000 worth of savings, that means someone with a hundred million dollars has 1,000 times more money than you. No doubt, 1,000 is way easier to wrap your mind around than a hundred million.

This technique can also be used to assess risk.

“If you go hang gliding, you have a roughly 1 in 116,000 chance of being killed during that flight,” he says. “Is that a lot of risk? It’s very tough to tell. But here’s another way to think about it. If you’re a 30 year old male in the U.S., you have about a 1 in 260,000 chance of dying tomorrow. So that means that tomorrow, by going hang gliding once, you’re taking on 3.2 times more risk than you usually do in a given day! So that gives a new way of thinking about hang gliding risk if you’re a 30 year old male in the United States: You’re tripling your usual risk of death for each such flight you take.”

Again, like the previous trick, just be sure you really understand those familiar units as well as you think.

Incorporate Time

Another great idea suggested by Greenberg is to incorporate time to turn numbers into occurrences over time.

So, let’s take a figure like \$400,000,000 dollars — which happens to be Powerball’s next jackpot amount. How much money is that, really? (image: Mark LaMoyne/Shutterstock)

“That’s tough to make sense of,” says Greenberg. “But if you live for 60 more years, that’s 525,600 hours remaining in your life, so if you win that jackpot — not taking into account time/value discounting and inflation considerations — that’s like getting paid \$761 per hour for each hour in the rest of your life (including when you’re asleep).”

Here’s another example: San Francisco’s metro area has about 4.3 million people. How many is that? Well, if you spoke to each person for one minute, and you did that eight hours a day, it would take you 24.5 years to speak to them all. Whoa.

“To me, that gives me the sense of how insanely unknowable a community of that size is,” adds Greenberg.

This reminds me of a tactic used by the Soviets at the Battle of Stalingrad during World War II. To demoralize the enemy, loudspeakers blared this rather discouraging message: “Every seven seconds a German soldier dies in Russia.” In reality, it was closer to nine each minute. That’s about 540 deaths an hour, 12,960 a day, 90,720 a week, or 388,800 a month.

There’s also the Battle of Borodino to consider — an incredibly bloody 19th Century engagement in which some 65,000 soldiers were killed over an eight hour period. To better conceptualize this, historian Gwynne Dyer used a trick we described earlier, the batching method. He compared the carnage at Borodino to “a fully-loaded 747 crashing, with no survivors, every 5 minutes for eight hours.”

Avoid Exponents

I asked Greenberg if exponents, or scientific notation, can be helpful. He said that scientific notation is fantastic for helping us calculate with large numbers — 10 billion divided by 1 million becomes 1010 / 106 = 10(10-6) = 104 — but it is not very helpful for grasping them intuitively. In fact, for intuitions sake, he says it can be confusing.

“Look at the two numbers, 1032 and 1039,” he says. “It’s easy to forget that the latter is TEN-MILLION times bigger than the former, since written in this notation they seem to be off by a “mere” 7 (i.e. 39-32 = 7).”

Hitting a Wall

Needless to say, there are limits to these tricks. We can only manipulate these figures so much before they once again regress into meaninglessness. For example, I asked Greenberg how we could better conceptualize something as large as the total number of stars in the Milky Way, which has about 300 billion stars.

Serge Brunier

“That’s really a tough one,” he replied. “The best I can do off the cuff is think about it in terms of the human population. If we somehow could colonize the Milky Way, that would be 300 billion / 7.1 billion = about 42 stars for each person currently alive on the earth today. Not super satisfying, but maybe that helps a bit. Of course, the number of people on the earth today is so large that it’s hard to make sense of even that number.”

Ultimately, Greenberg says large numbers will remain beyond our grasp (at least until the era of human cognitive enhancement), especially the really huge ones, and ones that we can’t relate to meaningful things we already understand.

“This is why when dealing with large numbers we often have to just ‘do the math’,” he says, “rather than trying to truly grasp these numbers. That is, we can work with extremely large numbers in calculations, and still get useful answers out the other end, we just can’t necessarily grasp the numbers used throughout that process.”

Top image: NASA.

George is a senior staff reporter at Gizmodo.

DISCUSSION

DiallingWand

"So, you could envision 200,000 people as 20 hockey arenas worth of people."

That should say 400,000.

"So, let's take a figure like \$400,000 dollars — which happens to be Powerball's next jackpot amount. How much money is that, really?

"That's tough to make sense of," says Greenberg. "But if you live for 60 more years, that's 525,600 hours remaining in your life, so if you win that jackpot — not taking into account time/value discounting and inflation considerations — that's like getting paid \$761 per hour for each hour in the rest of your life (including when you're asleep)."

And I think that should say \$400,000,000.

These numbers are pretty hard to grasp methinks. :P