Chapter XVI: Appendix: III

ON SCALES OF NOTATION.

We are so well accustomed to 10, 100, &c., as standing for ten, ten tens, &c., that we are not apt to remember that there is no reason why 10 might not stand for five, 100 for five fives, &c., or for twelve, twelve twelves, &c. Because we invent different columns of numbers, and let units in the different columns stand for collections of the units in the preceding columns, we are not therefore bound to allow of no collections except in tens.

If 10 stood for 2, that is, if every column had its unit double of the unit in the column on the right, what we now represent by 1, 2, 3, 4, 5, 6, &c., would be represented by 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, &c. This is the _binary_ scale. If we take the _ternary_ scale, in which 10 stands for 3, we have 1, 2, 10, 11, 12, 20, 21, 22, 100, 101, 102, 110, &c. In the _quinary_ scale, in which 10 is five, 234 stands for 2 twenty-fives, 3 fives, and 4, or sixty-nine. If we take the _duodenary_ scale, in which 10 is twelve, we must invent new symbols for ten and eleven, because 10 and 11 now stand for twelve and thirteen; use the letters _t_ and _e_. Then 176 means 1 twelve-twelves, 7 twelves, and 6, or two hundred and thirty-four; and 1_te_ means two hundred and seventy-five.

The number which 10 stands for is called the _radix_ of the _scale of notation_. To change a number from one scale into another, divide the number, written as in the first scale, by the number which is to be the radix of the new scale; repeat this division again and again, and the remainders are the digits required. For example, what, in the quinary scale, is that number which, in the decimal scale, is 17036?

5)17036
-----
5)3407 Remʳ. 1
----
5)681 2
---
5)136 1
---
5)27 1
--
5)5 2
-
5)1 0
-
0 1

_Answer_ 1021121

Quinary. Decimal.
_Verification_, 1000000 means 15625
20000 1250
1000 125
100 25
20 10
1 1
------ -----
1021121 17036

The reason of this rule is easy. Our process of division is nothing but telling off 17036 into 3407 fives and 1 over; we then find 3407 fives to be 681 fives of fives and 2 _fives_ over. Next we form 681 fives of fives into 136 fives of fives of fives and 1 five of fives over; and so on.

It is a useful exercise to multiply and divide numbers represented in other scales of notation than the common or decimal one. The rules are in all respects the same for all systems, _the number carried being always the radix of the system_. Thus, in the quinary system we carry fives instead of tens. I now give an example of multiplication and division:

Quinary. Decimal.
42143 means 2798
1234 194
------ -----
324232 11192
232034 25182
134341 2798
42143
--------- ------
114332222 542812

Duodecimal. Decimal.
4_t_9)76_t_4_e_08(16687 705)22610744(32071
4_t_9 1460
----- 5074
2814 1394
2546 689
-----
28_te_
2546
------
3650
3320
-----
3308
2_t_33
------
495

Another way of turning a number from one scale into another is as follows: Multiply the first digit by the _old_ radix _in the new scale_, and add the next digit; multiply the result again by the old radix in the new scale, and take in the next digit, and so on to the end, always using the radix of the scale you want to leave, and the notation of the scale you want to end in.

Thus, suppose it required to turn 16687 (duodecimal) into the decimal scale, and 16432 (septenary) into the quaternary scale:

16687 16432
Duodecimals into Decimals. Septenaries into Quaternaries.
1 × 12 + 6 = 18 1 × 7 + 6 = 31
× 12 + 6 × 7 + 4
--- ----
222 1133
× 12 + 8 × 7 + 3
---- -----
2672 22130
× 12 + 7 × 7 + 2
------ -------
_Answer_ 32071 1021012

Owing to our division of a foot into 12 equal parts, the duodecimal scale often becomes very convenient. Let the square foot be also divided into 12 parts, each part is 12 square inches, and the 12th of the 12th is one square inch. Suppose, now, that the two sides of an oblong piece of ground are 176 feet 9 inches 7-12ths of an inch, and 65 feet 11 inches 5-12ths of an inch. Using the duodecimal scale, and _duodecimal fractions_, these numbers are 128·97 and 55·_e_5. Their product, the number of square feet required, is thus found:

128·97
55·_e_5
---------
617_ee_
116095
617_ee_
617_ee_
------------
68_e_8144_e_

_Answer_, 68_e_8·144_e_ (duod.) square feet, or 11660 square feet 16 square inches ⁴/₁₂ and ¹¹/₁₄₄ of a square inch.

It would, however, be exact enough to allow 2-hundredths of a foot for every quarter of an inch, an additional hundredth for every 3 inches,[58] and 1-hundredth more if there be a 12th or 2-12ths above the quarter of an inch. Thus, 9⁷/₁₂ inches should be ·76 + ·03 + ·01, or ·80, and 11⁵/₁₂ would be ·95; and the preceding might then be found decimally as 176·8 × 65·95 as 11659·96 square feet, near enough for every practical purpose.

[58] And at discretion one hundredth more for a large fraction of three inches.