#264 - Maths: Dimensional Analysis

5 minute read

Motto: Knowledge = Voltage × Current!

This column is going to be fun, but first you've got to earn that fun through some learning. I'm going to start this post off with a lesson about one of the most useful-in-the-real-world mathematical concepts I ever learned. Today, we're going to look at dimensional analysis.

The word "dimension" has several different definitions. When it comes to math and science, I tend to think of "dimensions" as all the independent variables of a problem. In other words, all of the things you can change without changing anything else.

Easy example: Longitude, Latitude, Altitude
You if you're on an elevator, you're changing your altitude without changing your longitude and altitude. Altitude is a different dimension than the other two. Similarly, you could walk directly towards the North Pole and change your latitude without changing your longitude. These are dimensions.

Dimensions are measured in units. Engineering (or, perhaps, science in general) is math + units. When you introduce units into math, you transcend the abstract. 3 bananas + 2 bananas = 5 bananas. I like that better than 3 + 2 = 5. Often times, in real world problems, you can utilize units to direct your math.

You travel 100 miles 5 hours. What was your average speed?

Speed is length divided by time (meters/seconds or miles/hours).
So, to figure out speed, we see that we have to divide miles by hours. 100 {miles} / 5 {hours} = 20 {miles/hour}.

But what if the question was this:
You travel 528000 feet in 300 minutes. What was your average speed?

Well, it's technically a correct answer to say 528000 {feet} / 300 {minutes} = 1760 {feet/minute}... but that's not really useful. We want to get our answer in units we can understand.

We can get around this by utilizing a trivial mathematical concept, applied in a clever way.

These two things are true, regardless of what "A" is:
A × 1 = A
A / A = 1

You can multiply anything by 1, and still have that thing. You can divide anything by itself and get 1.

Here's the tricky part:
60 {minutes} = 1 {hour}
5280 {feet} = 1 {mile}

(60 {minutes} / 1 {hour}) = 1
(1 {mile} / 5280 {feet}) = 1

So, you can multiply anything by (60 {minutes} / 1 {hour}) and still get that thing (just expressed in terms of new units). You can do the same thing with (1 {mile} / 5280 {feet}).

Applying this A × 1 = A concept, utilizing the fact that (60 {minutes} / 1 {hour}) = (5280 {feet} / 1 {mile}) = 1, we can can solve our problem:

(1760 {feet/minute}) × 1 × 1
= (1760 {feet/minute}) × (60 {minutes} / 1 {hour}) × (1 {mile} / 5280 {feet})
= (1760 × 60 {feet × minutes × miles}) / (5280 {minutes × hours × feet})
= (105600 {feet × minutes × miles}) / (5280 {minutes × hours × feet})
= 20 {miles/hour}

Dimensional analysis is basically just the process of making the units work out (or, checking if your process ends with the correct units). You convert units by cleverly mulitplying by 1 many times in many ways. You can extend this concept into much more difficult problems.

If the first question was so simple you just said "well this isn't useful, I could just do that in my head". Then this harder is for you:

If a person drinks X liters of water per day, and you have Y people stranded on an island, for how many weeks (Z) can the people stay hydrated on the island (in terms of X & Y)?

We have X, measured in {liters/(days × persons)} and Y, measured in {persons}. We need Z, measured in {weeks}. We know 1 week = 7 days, so we can do unit conversion easily.

Z {weeks} = 1/7XY {liters/(day*persons)} 7 {days/week} × Y {persons} = 7XYZ {liters}

In school and in real life I have done tons of things like this. You'll see some fake/fun examples in a second.


Here is some fun with dimensional analysis:

House of Cards:

  • I estimate that 1 out of every 8 houses in the United States has a deck. 
  • There are 52 cards in a standard deck.
    • 1 {deck} / 8 {houses} × 52 {cards} / 1 {deck} = 6.5 {cards/house}
  • Thus - a house of cards must be made from an average of 6.5 cards.

This is hard. I'm going to do one more and get over it. I've tried SEVERAL (see the Top 5) that haven't worked out so well.

This one is a little more strict - this equality does actually work mathematically - even if it's meaningless:
For this post I took great inspiration from the author of XKCD, who has basically done a more accurate version of this exact same thing before:  

Top 5: Dimensions I Tried Forcing
5. Dollars/mouth (saw blades/tooth × $/blade movies × teeth/mouth)
4. Bacon packs/pizza (bacon packs/slices 
× slices/pizza)
3. Artists/cave (hits/artist 
× bats/hits × cave/bats)
2. Chassis/bankruptcy wedges (chassis/blade 
× blades/fusion × fusion/wheels × Wheel of Fortune wheel/bankruptcy wedges)
1. Dollars/Kitkat ($/commercial 
× commercials/break × breaks/Kitkat)

"I was like 'guys, excuse me I'm just gunna do some pushups and scream for a minute'."
- Josh, while describing an exciting experience he had -