For example, look at two businesses:

• Annie’s Art Gallery sells a single, \$1000 piece every day
• Frank’s Fish Emporium sells 250 trout at \$4/each every day

By the numbers, they’re identical \$1000/day operations, right? Hah.

Here’s how each business actually behaves:

Transactions are the workhorse that drive income, but they’re lost in the dollars/day description. When studying an idea, separate the results into Oomph and Often:

With Oomph and Often, I visualize two distinct levers to increase. A ratio like dollars/day makes me stumble through thoughts like: “For better results, I need 1/day to improve… which means the day gets shorter… How’s that possible? Oh, that must be the portion of the day used for each transaction…”.

Why make it difficult? Rewrite the ratio to include the root case: What’s the Oomph, and how Often does it happen?

Horsepower, Torque, RPM

In physics, we define everyday concepts like “power” with a formal ratio:

Ok. Power can be explained by a ratio, but we’re already in inverted-thinking mode. Just another hassle when exploring an already-tricky concept.

How about this:

Easier, I think. What could Oomph and Often mean?

Well, Oomph is probably the work we do (such as moving a weight) and Often is how frequently we do it (how many reps did you put in?).

In the same minute, suppose Frank lifted 100lbs ten times, while Annie lifted 1000lbs once. From the equation, they have the same power (though to be honest, I’m more frightened by Annie.)

An engine mechanic might internalize power like this:

What does that mean?

• Torque is the Oomph, or how much weight (and how far) can be moved by a turn of the engine (i.e., moving 500lbs by 1 foot)

• RPM (revolutions per minute) is how frequently the engine turns

A motorcycle engine is designed for reps, i.e. spinning the wheels quickly. It doesn’t need much torque — just enough to pull itself and a few passengers — but it needs to send that to the wheels again and again.

A bulldozer is designed for “Oomph”, such as knocking over a wall. We don’t need to tap into that work very frequently, as one destroyed wall per minute is great, thanks.

I’m not a physicist or car guy, but I can at least conceptualize the tradeoffs with the Oomph/Often metaphor.

Gears can change the tradeoff between Oomph and Often in a given engine. If you’re going uphill, fighting gravity, what do you want more of? If you’re cruising on a highway? Trying to start from a standstill? Driving over slippery snow? Lost the brakes and need to slow down the car?

Oomph/Often gets me thinking intuitively, Work/Time does not.

Variation: Electric Power

Electric power has the same ratio as mechanical power:

Yikes. It’s not clear what this means. How about:

It’s hard to have ideas out of the blue, but we might imagine something (a mini-engine?) is moving the Oomph around inside the wire. If we call it a “charge” then we have:

And we can give those subparts formal names:

• Voltage (Oomph): How much work each charge contributes

• Current (Often): How quickly charges are moving through the wire

Now we get the familiar:

Boomshakala! I don’t have a good intuition for electricity, at least my goal is clear: find analogies where voltage means Oomph, and current means Often.

And still, we can take a crack at intuitive thinking: when you get zapped by a doorknob in winter, was that Oomph or Often? What attribute should batteries maximize? What’s better for moving energy through stubborn power lines? (Vive la résistance!)

The ratios think every type of power reduces to a generic Work/Time calculation. The Oomph/Often metaphor gets us thinking about Torque/RPM in one scenario and Voltage/Current in another.

What’s Really Going On? Parameters, Baby.

The Oomph/Often viewpoint lets us think about the true cause of the ratio. Instead of dollars and days, we wonder how the actual transactions affect the outcome:

• Can we increase the size of each transaction?

• Can we increase the number each day?

In formal terms, we’ve introduced a new parameter to explain the interaction. To change a ratio from a/b to one parameterized by x, we can do:

We change our viewpoint to see x as the key component. In math, we often switch viewpoints to simplify problems:

• Instead of asking what happens to the observer, can we change parameters and ask what the mover sees? (Degrees vs. radians.)

• Can we see a giant function as being parameterized by smaller ones? (See the chain rule.)

• Can we express probabilities as odds, instead of percentages? (It makes Bayes Theorem easier.)

Adjusting parameters is a way to morph an idea that doesn’t click into one that does. Since I don’t naturally think with inverted units, I’ve made it easier on myself: deal with two multiplications, instead of a division.

Happy math.

• 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:

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.

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

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 (a vacation)
• 1 billion seconds is 30 years (a career)
• 1 trillion seconds is 30,000 years (longer than human civilization)

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 millimeter 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 approach 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.

Time and Distance

60 mph = 1 mile per minute

• Going 60 mph and the exit is in 10 miles? That’s 10 minutes.
• Been driving a half hour? That’s about 30 miles at highway speeds.

Feet Per Second = MPH * 1.5

MPH = Feet Per Second * 2/3 (derivation)

• 60 mph is about 90 feet per second (88 exactly), so just multiply by 1.5. Or, just add half to itself (60 + 30 = 90).
• Going 100 mph? That’s 150 fps.
• Going 10 fps? That’s about 7 mph (10 * 2/3 is 6.666). Or, just take away 1/3 (10 – 3 = 7).

speed of light = 1 foot per nanosecond (derivation)

• The US is about 3000 miles long. There’s about 5000 feet/mile, so that’s about 3000 × 5000 or 15 million feet. 15 million feet takes 15 million nanoseconds, or 15/1000, or 15 milliseconds. That’s the minimum time for a signal to go across the country.
• Inside a microchip, if you have a clock cycle every nanosecond (1 GHz), your signal can only travel 1 foot at most (or less, depending on the material). Even light takes 30ns to cross a 30 foot room.

1 year = 250 work days = 2000 work hours (derivation)

• Project takes 1000 man hours? That’s 6 months for 1 person.
• Daily commute of 1/2 hour? That’s .5 * 250 = 125 hours in the car each year.

Money and Finance

• \$1/hour = \$2000/year** (derivation)

• Earn \$25/hour? That’s about 50k/year.

• Make 200k/year? That’s about \$100/hour. This assumes a 40-hour work week.

• \$20/week = \$1000/year** (derivation)

• Spend \$20/week at Starbucks? That’s a cool grand a year.

Rule of 72: Years To Double = 72/Interest Rate (derivation)

• Have an investment growing at 10% interest? It will double in 7.2 years.
• Want your investment to double in 5 years? You need 72/5 or about 15% interest.
• Growing at 2% a week? You’ll double in 72/2 or 36 weeks. You can use this rule for any duration of time, not just years.
• Inflation at 4%? It will halve your money in 72/4 or 18 years.

Mental Arithmetic

Numbers

10,000 = hundred hundred

million = thousand thousand

billion = thousand million

trillion = million million

• 1% of 10k is 100. The Dow is roughly 10k (it’s about 12k now). So if the dow drops 100, it’s about a 1% loss.
• What’s 5k x 50k? That’s 250 * thousand * thousand or 250 million.

Visualizing numbers (read more)

• 12 days = 1 million seconds
• 1 year = 31 million seconds (about pi * 10 million)
• 30 years = 1 billion seconds

• 30,000 years = 1 trillion seconds

• One “part per million” means an accuracy of 1 second every 12 days. One “part per trillion” means an accuracy of 1 second every 30,000 years.

Powers of 2

2^6 = 64 (the sixes match: six and sixty-four)

2^10 ~ thousand (1 kb)

2^20 ~ million (1 mb)

2^30 ~ billion (1 gb)

• Sure, 2 to the tenth = 1024, but 1000 is good enough for government work. (Read on about KB vs KiB).
• Have 32-bit color? That’s 2 + 30 bits = 2^2 * 2^30 = 2^2 billion = 4 billion (4gb exactly).
• Have a 16-bit number? That’s 6 + 10 bits, or 2^6 thousand, or 64 thousand (64 kb).

a% of b = b% of a

• It’s not immediately clear, but it’s true: a% of b = .01 * a * b, which is the same as b% of a (.01 * b * a).
• What’s 16% of 25? The same as 25% of 16: 4
• What’s 43% of 200? Same as 200% of 43: 86.