Imperial Base v2

Envious of metric systems?

In the metric system you can just say 1000m = 1km and so on. Did you know that this is actually because the decimal notation is rigged in the metric system's favour!

Version 2 introduces chains and furlongs instead of factoring one mile as 11×16×10.

Introducing Imperial Base v2 (fractions of inches)

Time to use a number base that's rigged for imperial. Let inches be the 1 in our system. Inches subdivide into sixteenths, so anything after the inch symbol is going to be base 16 (0-9 plus A-F)

Value Written as
3/16 inch 0″3
15/16 inch 0″F
17/256 inch 0″11
255/256 inch 0″FF

Note, that the inch symbol (double prime: ″) serves as a kind of decimal point. But of course, not decimal because that's base 10. And the same way each digit after a decimal point is multiplied by 10-1 (or 1/10), 10-2 (or 1/100), 10-3 and so on.

Introducing Imperial Base v2 (inches)

Moving onto whole inches, there is 12 inches in one feet, so the inches digit is going to be base 12 (0-9 plus A, B).

Value Written as
12 inches 10″
15 inches 13″
35 inches 2B″

Note that 13″ is not a valid representation. It is a valid value, but it is represented as 11″ (1 feet and 1 inch). And inches go up to B not up to 9, so your digits are 0123456789AB.

Introducing Imperial Base v2 (feet)

And three feet make one yard, so this digit is base 3 (0-2).

Value Written as
37 inches 101″
4 feet 110″
5 feet 120″
6 feet 200″

Again, note that 30″ is not a valid representation.

Introducing Imperial Base v2 (yards)

Next 1760 yards make up one mile. We don't have that many choices for digits, but fortunately 22 yards also make up one chain! How convenient this imperial system is, we happen to have 22 digits in the form of 0-9 then A-L! 22 yards in a chain 10 chains in a furlong 8 furlongs in a mile

Value Written as
1 chain 1 000″
23 yards 1 100″
2 chains 2 000″

Introducing Imperial Base v2 (chains)

Then 10 chains make up one furlong. What's that, base 10? Feels weird.

Value Written as
1 furlong 10 000″
1 furlong minus 1 chain 9 000″

Introducing Imperial Base v2 (furlongs)

Then 8 furlongs make up a mile!

Value Written as
1 mile minus 1 inch 79 L2B″
1 mile plus 1 inch 100 001″

Introducing Imperial Base v2 (miles)

Miles then could just be represented in base 10, but that feels too metric. So how about base 36, that combines the base 12 of inches and the base 3 of feet, and can be represented using all the numbers and letters (0-9 plus A-Z).

Value Written as
40, 450 yards MAD 000″
28, 101 miles LOL00 000″
Earth's equatorial circumference (24,901.461 miles) J7 P36 J10″F5C 29
Earth's polar circumference (24,859.734 miles) J6 J58 F26″3D7 0A4

Calculator v2

Value Written as
36 miles minus 17/256 inch ZAF 92B″EF
=

Note, multiplication and division might not make sense, in that because the result is in a different unit, it would need its own number system, say Imperial² Base. This only applies when you are multiplying two dimensions together, multiplying a dimension by a scalar makes sense (in which case units should be left as inches for the scalar).