How to Develop a Sense of Scale

Get the Math, Better Explained eBook and turn Huh? to Aha!

A sense of scale helps us better understand the world, and convey ideas more effectively. What’s more impressive?

  • Bill Gates has 56 billion dollars.
  • Bill Gates earned over $3000 per minute [$50/second] since Microsoft was created. Spending 5 seconds to pick $100 off the floor is literally not a good use of his time.

If you’re like me, the second statement makes your jaw drop. 56 billion is just another large number, but $3000 per minute is something vivid and “imaginable”. Let’s check out a few ways to convey a sense of scale.

Compare Side By Side By Side

A common way to put things in perspective is to literally line them up, side by side. We’re visual creatures. We like to see, not imagine abstract numbers. To our brains, a million, billion, and trillion all seem like large, vague numbers.

Apple knows this. Many of its ads compare products to everyday objects, rather than touting the raw dimensions:

Apple ads relative size

The Macbook Air fits into a manilla envelope. The ipod nano is as thick as a pencil. Certain cameras fit in a box of altoids. You know their size without busting out a ruler. Just yesterday, I got a haircut with the #5 clippers (“As wide as your finger”) and knew what it meant. The hairdresser didn’t have to say “.875 inches”.

It seems backwards that “casual” measurements like a pencil’s width can be more useful than a count of millimeters. But we’re not machines — our everyday experience is with pencils, not millimeters, and we can easily imagine how much room a pencil takes.

Here’s a few more examples of side-by-side comparison in action — notice how well they convey a sense of scale.

Relative size of planets & stars. A great example, much preferred to “Boys and girls, the Sun’s diameter is 1000x larger than the Earth’s”.

Relative Dimensions of Fictional Ships & Characters. Fun and interesting: occupy a geek for hours by asking how many TIE fighters would be needed to take out the Starship Enterprise.

Relative ship sizes

Interactive Sense of Scale Flash App. A fantastic way to visualize the relative sizes of objects.

Relative ship sizes

And of course, the famous power of ten video:

Rescale and Resize

Instead of looking up at the “big numbers”, we can shrink them to our level. Imagine the average person makes 50k/year, and a rich guy makes 500k/year. What’s the difference?

Well, instead of visualizing having 10x your money, imagine that things cost 10 times less. A new laptop? That’ll be 150 bucks. A new porsche? Only 6,000 dollars. A really nice house? 50k. Yowza. Things are cheap when you’re rich.

To understand Bill Gates’ scale, don’t think of 50 billion dollars and 5 billion/year income — it’s just another large number. Try to imagine having things cost 100,000 times less (and 100,000 is a pretty large number).

A laptop would be a few pennies. A porsche would be about 60 cents. Your $50M mansion would be a mere 500 bucks. You could “splurge”, spend $1000, and get everything you’ve ever needed. And you’re still earning 50k/year.

It’s much more vivid than “50 billion in the bank”, eh?

Use What We Know: Time and Distance

Sometimes, a different type of scale may be useful. We know time and distance, which cover a surprisingly broad range of sizes.

For most of us (myself included), millions, billions and trillions are “big”. It’s not intuitively obvious that a trillion is actually a million squared — that is, a trillion makes 1 million look imperceptible.

Check out these brain-bending figures:

  • 1 second is 1 second
  • 1 million seconds is 12 days (Interesting)
  • 1 billion seconds is 30 years (Wow, that’s a lot)
  • 1 trillion seconds is 30,000 years (Jumpin’ Jillikers!)

Yowza. Do you feel the staggering difference between a trillion and a million? Between a billion and a million?

We get a similiar effect when thinking about distance:

  • 1 millimiter is 1 mm (pretty tiny)
  • 1 million mm is a kilometer (down the street)
  • 1 billion mm is a 1000 km (600 miles — partway across the country)
  • 1 trillion mm is 1,000,000 km (Going around the world 25 times, almost as wide as the Sun)

Again, see the difference? How small a million is (“down the street”) compared to the size of the Sun?

These numbers come in handy in many applications:

  • 99.999% reliability (“Five 9′s”) means an error rate of 10 out of a million. That is, you can be offline for only 10 seconds every 12 days. Or, you can have a tolerance of 10mm for every kilometer. That’s pretty accurate!
  • “One part per million” is often used by chemists to measure concentrations of substances. One ppm is like having a presence of 1 second in 12 days. And a part per trillion? You got it: 1 second every 30,000 years. That’s tiny.

This apprach helped me understand how utterly gigantic a trillion is, and how precise 99.999% really is.

Use People, Places and Things

Yet another approach is to combine things we’re familiar with. Here’s a few numbers:

  • There’s about 6.5 billion people on Earth
  • The internet has many billions of pages (call it a trillion to be safe)

The US deficit of 10 trillion dollars would require a tax of $10 for every page on the internet to pay off (Yowza! And these are with generous estimates of the internet’s size).

A GUID, or large ID number used in programming, is at no risk of running out. How many are there? Well, we could give everyone a copy of the internet, every second, for a billion years… and still have enough GUIDs to identify each page. See how much bigger that is than “2^128″? (For the geeks: yes, the birthday paradox makes the chance of collision much higher).

Seeing a number impact the real world (i.e. being applied to every page of the internet) makes an idea come to life.

Summary

This article isn’t really about numbers. It’s about understanding and communication, how we think and convey ideas. Do you insist on rigid scientific terms, or do you reach out to your audience with terms they understand? Do you think a “lay person” (someone who happened to choose a different field of study than you) is more interested in raw numbers, or side-by-side demonstrations?

Developing a sense of scale helps us better understand the world and better convey that understanding.

In a perfect universe, we’d hear “one trillion”, imagine a million by million grid, and say “wow”. But that’s not the case — in order to say “Wow!” we need (or at least I need) to imagine the number of seconds in 30,000 years, longer than modern human civilization.

When presenting ideas, remember that analogies can be more powerful, interesting and effective than a 1 with 12 zeros.

Other Posts In This Series

  1. Rethinking Arithmetic: A Visual Guide
  2. Techniques for Adding the Numbers 1 to 100
  3. Understand Ratios with "Oomph" and "Often"
  4. Mental Math Shortcuts
  5. How to Develop a Sense of Scale
  6. Easy Permutations and Combinations
  7. How To Understand Combinations Using Multiplication
  8. Surprising Patterns in the Square Numbers (1, 4, 9, 16…)
  9. Navigate a Grid Using Combinations And Permutations
  10. A Quick Intuition For Parametric Equations
  11. Understanding Algebra: Why do we factor equations?
  12. How To Learn Trigonometry Intuitively
Better Explained is dedicated to helping you grasp concepts, and serves over 250k readers each month.

Enjoy this article? Try the site guide or join the newsletter:

30 Comments

  1. Great article! The scaling down of prices to reflect huge incomes *does* bring them home in a new way.

    (BTW, working on a High-Availability application, I am painfully familiar with the time implications of 5-nines )

  2. Thanks, glad you liked it! Yeah, fortunately I haven’t had to be on the business end of a 99.999% application :).

  3. wow, it makes it much easier to relate to wealth when we reduce how much things cost to them! a porshe for 6000! a A macbook air for 179.99!!!

  4. Appreciate the comment Ankur — yeah, it’s a fun way to look at it. Although in reality, I’m sure Bill could pick up money & strategize at the same time :).

  5. Great article, as usual, though would like to point out that in finance a trillion is 1,000 million (not a million million as it logically should be), so the US deficit is not quite as large as you suggest.

  6. @Alex: No worries, in the UK they often have different terms for billion.

    @Simpsons: Thanks for the info, glad you liked the article. I love the simpsons :).

  7. That makes my brain hurt a little bit thinking about all of those things and sizes and how they compare but it all really makes sense. All of my thanks go out to bob, the ingenious creator of this fantastic site as well as partly the author of this amazingly brain hurting article. Bob, i love you!!!
    and by the way i really learned a lot from this article and i will now think of the world in a totally different way than i thought of it ten minutes ago when i was mindlessly playing action games on brainbashers.com and not caring about size and humans compared to dinosaurs and the earth compared to some extremely huge random star thing way out in the universe. GO BOB!!!

  8. @Hayley: Thanks for the comment — a lot of these ideas hurt my brain too :).

    @Techdudes: Thanks, glad you liked it.

  9. This was an amazing journey into the realm of scale and dimension! Thanks for leading me into such a fascinating new realm of perception.

    This sort of thing really broadens my perspective and expands my view by a tenfold, if not by a magnitude of one trillion.

    What I find even more synchronistic about encountering your post here is that a family member just mentioned to me a couple of weeks ago that she could not intuitively grasp the reality of Bill Gates’ net worth. I’m going to suggest this article to her, haha.

  10. Hi Jeffrey, glad you enjoyed the post! I was similar — it’s hard to wrap our heads around how an extremely wealthy person sees the world. Having expensive things cost pennies helps bring it to life, I think.

  11. Hi Khalid, Just like that I crossed your site and it made me to park and browse through all your article. Simply fantasic man fantastic.

    I could see that you use simple tool ( guess Word, Powerpoint ) to make shiny graphics.

    I loved your articles!!! :-)

  12. Loved the article.

    It reminded me of a chemistry professor who tried to get us to wrap our heads around the number of atoms in a mole (6.023 X 10^23) by demonstrating that a mole of M&M’s would completely cover the earth (although I don’t remember the depth, it could have been 3ft or 3 miles). Either way, it was a pretty effective image.

  13. Thanks Led — that’s a really nice example too! When numbers get to that scale, we need to find ways to relate it to everyday objects (the earth) and not just have some giant exponent.

    Just for fun, we can do a quick calculation assuming a m&m is about 1cm x 1cm x 1cm. I figure you could make a layer about 4000 feet high! (http://tinyurl.com/4fs23g). Thanks for the comment.

  14. I love the site. I’m really impressed by demonstrating the net worth of the super-rich on a distance scale.

    Imagine asking a classroom of students to make a bar graph of three individuals’ net worth: an average Joe ($100k), Biff the millionaire($1 mil), and a Warren Buffett or Bill Gates (prior to the fall in the stock market–$65 bil).

    If the bar graph’s scale is 1 inch: $1 million, then Average Joe’s height is 1/10 inch which is much smaller than Biff’s 1 inch. But they’re both DWARFED by the Buffett/Gates wealth, which at this scale would have a height of over 1 mile!!!

  15. There is an re-make of the Powers Of Ten movie, “Cosmic Voyage”, narrated by Morgan Freeman.
    It is available in 720p on bittorrent.

  16. @Greg: Thanks for the comment! Wow, that’s really amazing, I like that visualization a lot — it’s amazing how much of a difference that is, an inch vs a mile!

    @dt: Cool, thanks for the info.

  17. reminds me of a quote by the president of coca-cola once, something along these lines: “a billion years ago life emerged. a billion minutes ago christianity was getting started. a billion cokes ago was yesterday.”

  18. Hi, I just discovered this site and I love it!
    This latest post is really great. I did my undergrad in biology, where we often lose track of the scale factor. We learn how one molecule behaves, not even=r being reminded that the phenomenon occurs a gazillion times a second in a given cell…

  19. @Aaron: Thanks, that’s a great quote. I had no idea so much soda was consumed every day.

    @Scientific Chick: Thanks! Yeah, when you think about it, our body is a giant ecosystem of billions of cells, each with countless reactions taking place. The whole operation is quietly humming along with scant notice from us :).

  20. Although there still isn’t any intuitive explanation on big numbers like grahams number (g64). Even g1 is pretty unthinkable.

  21. Great article Kalid! It’s unbelievable to analyze the scale of these large numbers in a way that our minds can actually comprehend. The star comparison video that you used has been removed by the user. I’m not sure if this is the one you used, but this video is great if you are looking for a replacement: http://www.youtube.com/watch?v=HEheh1BH34Q

Your feedback is welcome -- leave a reply!

Your email address will not be published.

LaTeX: $$e=mc^2$$