Complementary numerals

Sometimes a change of notation can advance thought. In today's blog the tradition of numerals 6,7,8, and 9 will be challenged. 

Suppose the five letters a, b, c, d, and e are used to represent numbers satisfying

a + 1 = 0,  b + 2 = 0,  c + 3 = 0,  d + 4 = 0,  e + 5 = 0.

Then each numeral in {1,2,3,4,5} has a complementary numeral in {a,b,c,d,e}. If n is a numeral, a x n is the complementary numeral. For instance, a x a = 1. This equality is an instance of the "rule of signs", the others being a x 1 = a, and 1 x 1 = 1. The rule of signs and the principle of placeholder value are powerful principles of notation that enable modified decimal systems.

With modified numeration, the number five (5) can be written 10 + e = 1e, where e is in the ones place of a two-digit number. Six is written 1d, seven is 1c, eight 1b, and nine 1a. The old numerals for these numbers are no longer needed. Note that the decimal .5 is written 1.e so that if the value right of the decimal is lost (truncated), the number remaining is 1. , which is the round-off value of .5 . This property of truncation yielding round-off makes e-decimal notation of numbers more attractive than the usual decimal notation.

Suppose that {a,b,c,d} is the set of complementary numerals and 5 is used instead of 1e for five. In such a d-decimal system .5 rounds down to 0, unlike the usual rounding. Nevertheless, 1.d, 1.c, 1.b, and 1.a all round up to 1. on truncation. Without e, one has a x 5 = a5 = -10 + 5. To multiply a number of several digits by a, change every numeral to its complement, and when 5 is encountered, leave it 5 and add a to the column to the left.  Attentive students, such as read iMechanica, will take as an exercise the description of multiplication by a in the e-decimal system where 5 is not used. Evidently one of the costs of using complementary numerals is the extra complication in changing the sign of a multi-digit number.

Since four is twice two, and five is half of ten, products by two express those of four and five. Multiplication in the d-decimal system is reduced to two and three, once the rule of signs and products by a are in hand. The idea of numerals representing negative quantities is implicit in Roman numeral construction such as IX and XL representing 9 and 40 respectively. In 1726 John Colson published a description of arithmetic with “small numbers”, essentially the d-decimal system described above. In 1840 Augustin Cauchy illustrated use of this arithmetic by comparing the square of 11 with that of 1a, and the square of 12 with that of 1b. He also illustrated the repeating decimals of 1/7 and 1/73 using complementary numerals. For more references see the Wikipedia article Signed-digit representation.

Check that these physical constants agree with the ordinary decimal expressions:

c = 3.0a x 10^1b m/s (speed of light in vacuum)

G = 1c.cc x 10^aa m^3/kg s (gravitational constant)

N = 1d.02 x10^23 mol^a (Avagadro’s number)

R = 1b.31 J/mol K (gas constant)

eps= 1a.b5 x 10^ab C^2/N m^2 (permittivity of free space)

mp/me =2b40 (ratio of mass of proton to mass of electron)

g = 10.b1cd5 m/s^2 (acceleration of gravity at Earth’s surface)

Re = 1d.4c4 x 10^1d m (radius of the Earth)

Ms = 2.0aa x 10^30 kg (mass of the Sun)