Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to Be Called the Tonal System, With Sixteen to the Base

Description

Tonal (base-16) system, recommended by Luke-Jr

Decimal (base-10)

digits: 0-9
| Hexadecimal (base-16)
equivalent
digits: 0-9, A-F
| Decimal (base-10)

digits: 0-9 | Hexadecimal (base-16)
equivalent
digits: 0-9, A-F
---|---|---|---
1/2 = 0.5
| 1/2 = 0.8
| 11/16 = 0.6875
| B/10 = 0.B

1/4 = 0.25
| 1/4 = 0.4
| 13/16 = 0.8125
| D/10 = 0.D

1/8 = 0.125
| 1/8 = 0.2
| 15/16 = 0.9375
| F/10 = 0.F

3/4 = 0.75
| 3/4 = 0.C
| 1/32 = 0.03125
| 1/20 = 0.08

3/8 = 0.375
| 3/8 = 0.6
| 7/24 = 0.29166…
| 7/18 = 0.4AAA…

5/8 = 0.625
| 5/8 = 0.A
| 5/12 = 0.4166…
| 5/C = 0.6AAA…

7/8 = 0.875
| 7/8 = 0.E
| 1/3 = 0.3333…
| 1/3 = 0.5555…

1/16 = 0.0625
| 1/10 = 0.1
| 2/3 = 0.6666…
| 2/3 = 0.AAAA…

3/16 = 0.1875
| 3/10 = 0.3
| 1/6 = 0.1666…
| 1/6 = 0.2AAA…

5/16 = 0.3125
| 5/10 = 0.5
| 1/64 = 0.015625
| 1/40 = 0.04

7/16 = 0.4375
| 5/10 = 0.7
| 25/64 = 0.380625
| 19/40 = 0.64

9/16 = 0.5625
| 9/10 = 0.9
| 1/128 = 0.0078125
| 1/80 = 0.02

Look at how much simpler hex is for representing fractions! This is because 16 has more divisors than 10.
Programmers (and chefs, too!) deal almost exclusively in bases that are powers of 2 (like base 8, 16, 64, etc.).

This table comes from Table 3 (p. 21) of Nystrom's 1862 Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to Be Called the Tonal System, With Sixteen to the Base , modified to use conventional hex digits. So as to distinguish numerals from letters, Nystrom invents his own symbols for hex's 9 & B-F and uses the symbol "9" for hex's A.