At first glance, the US highway seems like a jumble of roads:
However, the numbers follow a grid, with rules nobody told me (image source, click to enlarge):
Even numbers go East/West (I-90, I-10), and odd numbers go North/South (I-5, I-95). Think "Even" goes "East".
Numbers increase towards the Northeast. (Hey, NYC thinks it's the center of the world, right?) I-5 is on the West coast, I-95 on the East coast. I-10 must be in Texas, I-90 must be in Massachusetts.
Auxiliary interstates connect to the primary ones, and have 3 digits: 290 connects to 90, 495 connects to 95, etc.
- Odd prefixes (190) connect once into the city from the interstate ("spur").
- Even prefixes (495) typically loop around a city. (Being a man-made system, there are exceptions.)
Whoa. There's so much information conveyed in a simple numbering scheme! Without looking at a map, I know I can drive from Seattle to Boston on I-90. Maybe I'll take I-95 South when I'm there and make my way to Florida. On the way I'll take I-10 West, over to LA, then drive up I-5 North back to Seattle.
How does this work?
We have a concept of a number, and all its properties (even/odd, size, number of digits...)
We noticed a real-world object (a highway) that had various properties (North/South, position, major/minor)
We associated the properties of the number to the properties of the object
This is thinking mathematically. It's not about doing arithmetic quickly, or memorizing formulas, it's about connecting patterns. Math is a zoo of made-up objects that we relate to ones in the real world. The "usefulness" of the made-up objects depends on our imagination.
Extending the highway system
Have we used all the interesting properties of a number? How about whether it's a prime number.
Suppose local routes used small prime numbers: Route 2, 3, 5, 7, 11. (Yep, remember that 2 is prime.)
Once the main routes are numbered, smaller roads that connect them can follow this rule:
If you connect two routes, use their product. 3 * 11 = 33, so Route 33 connects Route 3 and 11.
If you loop back to the same route, just square it. 3 * 3 = 9, so Route 9 connects Route 3 to itself.
If you connect three roads, it could be Route 66 (connecting routes 2, 3 and 11).
Will this always work? You bet. Any two primes, when multiplied, give a unique number. 33 will never be reached by any other combination of primes. (The fancy math phrase: every number has a unique prime factorization.)
Computer Science
See how we're trying to cram a bunch of information into a little number? That's the essence of binary data.
An eight-bit binary number like 01000100 is essentially eight true/false questions:
- Are you East/West? (1 if yes, 0 otherwise)
- Are you local connection? (1 if yes...)
- Are you a spur road?
- Treating your route number as a set of binary digits...
- Anything in the ones digit?
- Anything in the twos digit?
- Anything in the fours digit?
- Anything in the eights digit?
- Anything in the sixteens digit?
An 8-bit binary number can pack in a bunch of related questions into a single byte, and is what makes binary so efficient.
Onward and Upward
Numbers have a bunch of properties, right? Aren't we curious to discover more, like the remainder (modular arithmetic)? Maybe Route 12 (which is one set of 11, remainder 1) has some connection to Route 11.
Happy math.
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27 Comments on "The Math Inside the US Highway System"
You are sooo awesome; if I had teachers like you earlier in life, and later in HS, my path would be MUCH different! It’s clear you have a true passion and gift for teaching mathematics! I was always interested in the sciences from a young age, but my ambitions were killed by the horrible math teachers I had in the parochial system I was a victim of (I say victim b/c I was very young, grades 1-4; the nuns were ignorant of English! never mind mathematics. How the “school” was licensed by the state of MA I’ll never know). Later, the public HS I attended was not much different (I later realized it was run by political hacks..big surprise). My ability to do serious math was so severely damaged I lost ALL confidence in myself. It was many many years later that I realized I actually liked many aspects of math, but any chances at a “career” in scientific research (my interest from early on) were long gone. I write this as a warning to parents to be very very careful who you allow to “teach” your children, they can easily have their lives ruined by people who should never be allowed near young minds.
Thanks so much for the note. Yes, unfortunately we have several learning experiences (math is unfortunately the common one) where we think something is wrong with us, or we’re not capable of understanding something. The vast, vast majority of the time the confusion due to a poor explanation, and not a failing of the student. (Imaginary numbers are my favorite example here, most anyone can go from Huh? to Aha! with the right analogy.)
Really great article, Kalid, and which I’ve just shared on LinkedIn.
You’ll probably also be interested in the unique combination of just 3 words that identifies a 3mx3m square anywhere on the planet. It’s far more accurate than a postal address and it’s much easier to remember, use and share than a set of coordinates
(ref/quote: what3words.com).
Thanks Nigel! Neat project, I hadn’t heard of that before :).
Interesting post. Some more road tidbits: a 3 digit odd interstate is a spur, whereas an even 3 digit is a loop, that is it will return to the interstate. And on all two lane roads the mile markers will always increase if they are on your right, (at least in U.S. , maybe England is opposite?
Nice tidbits, thanks!
What I said of Spurs and loops apply to an interstate into a city.
Kalid,
Great post! Do you think there a mathematical pattern that applies to the NYC metro system? I have the hardest time making sense of the alphanumeric number system used for naming the NYC metro’s lines.
Hi Sachin, I don’t know NYC well enough, but I didn’t see a pattern to the labels after a quick search. It would have been useful to have an even/odd East/North distinction on the lines.
Or, to borrow from Marianne Moore, “An imaginary zoo with real animals in it. “
I like it :).
This is really interesting. I only ever knew the even/odd rule. Kalid, I really enjoy your newsletter and I am so glad I joined. You have a true gift for teaching and explaining math that is enlightening and refreshing. When I was young I always viewed math as numbers on paper but your explanations of calculus are so elegant that I feel I understand calculus far better and can relate it now to a broader set of experiences beyond formulas and memorization. Thanks. Happy math!
Thank you Roger! It’s fun sharing what helps things truly click with others. Really appreciate it.
Because of this article, my wife’s opinion that I am lost is poppycock! Kudos Kalid.
Hah, thanks :).
Hi Kalid, Very useful information. I like your way of teaching mathematics. Appreciate your effort. Keep doing this great work.
Very Tiny Comment!
2ed Pic look @ detail : I-90 goes through Cleveland N Buffalo city (Canada neighbors) is not there .
Yet dude you keep my mind sharpened, N read my thoughts, as if you were in my head!
Glad you liked it! Not sure about the accuracy of all the stops in the picture :).
I think the Bank’s account numbers works on the same principle….. ?
Yep, bank account numbers may have a similar structure (certain numbers to designate savings, checking, etc.). Not sure but I wouldn’t be surprised, it’s very useful to categorize items by the properties of the number.
Hi, I’m enjoying your articles from another part of the world. I’m living in Iran. our educational system is mostly based on memorizing stuff. but i’m trying to learn everything intuitively and with your ADEPT method. hopefully i know English and i can read all of the beautiful things you write here. Thank you for sharing all this :)
Crazy to think- your SSN# are formed with a similar formulas, crazy the world we live in!
Sadly it breaks horribly if we ever need to connect R-11 and R-3 again on the East side of that intersection… :(
Did you know that US highways (the white shields with black numbers) are the opposite of Interstates? The low numbers are in the east/north and the high numbers are in the west/south. US 1 hugs the Atlantic and 101 runs along the west coast. US 2 runs through the northern tier of states and US 80 went from Jacksonville to San Diego. The Interstate system was designed to be opposite to avoid confusion.
One constraint they wanted when they created the system: 76 had to go through Philadelphia. This took away lots of degrees of freedom
Not quite. Originally that was I-80 that went through Philly; they swapped portions of I-80 and I-76 in the 1970s
The most interesting part about the highway system, however, is not its human ordered grid, but I still adherence to fluid flow principles as its intended grid was neither maintained nor augmented as out progenitors had planned.
Water had an enormous almost juggernaut -like surface tension. In fact the surface tension is so strong that our best understanding of how water on the edges actually moves is diffusion
Next time you sit in traffic, remember this and move toward the center of the area of largest chaos–diffusion. They are your best bets at accomplishing smooth an natural, almost harmonic flow.