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.)
哇。在一个简单的编号方案中传递了如此多的信息!不用看地图，我知道我可以沿着I-90公路从西雅图开车到波士顿。也许我到那儿后会走95号州际公路，然后去佛罗里达。在路上，我将沿着10号州际公路西行，途经洛杉矶，然后沿着5号州际公路北上，回到西雅图。
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
Thisis 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 aprime number.
假设本地路由使用小素数:路由2、3、5、7、11。(Yep, remember that 2 is prime.)
Once the main routes are numbered, smaller roads thatconnectthem 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 auniquenumber. 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 ofbinary data.
An eight-bit binary number like01000100is essentially eight true/false questions:
- Are you East/West? (1 if yes, 0 otherwise)
- 你是本地的吗?(1)如果是…)
- Are you a spur road?
- Treating your route number as a set of binary digits...
- Anything in the ones digit?
- 有两位数的吗?
- Anything in the fours digit?
- Anything in the eights digit?
- Anything in the sixteens digit?
一个8位的二进制数字可以将一堆相关的问题打包到一个字节中，这是使二进制如此高效的原因。
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.