Notice that 2 ×2 = 4, and so it is appropriate to write
√
–
4 = 2. We also have that √
–
4 equals 4, 8, and 10, since
4 ×4 = 4, 8 ×8 = 4, and 10 ×10 = 4 in clock math. In
the same way, √
–
1 equals 1, 5, 7, or 11, and √
–
9 equals 3
or 9. There is no number equivalent to √
–
2, for instance,
in this system.
It is possible to give an interpretation of some frac-
tions in clock math. Consider 4/5 for instance. In ordi-
nary arithmetic, this fraction is a number xsuch that
5×x= 4. Looking at the fifth row of the table above,
we see that 5 ×8 = 4, and so it is appropriate to inter-
pret the fraction 4/5 as the number 8 in clock math. In
the same way, 1/5 = 5, 3/7 = 9, and 2/7 = 2. Notice,
however, that it is not possible to give an interpretation
to the fraction 1/6, for instance, since the number 1
does not appear anywhere in the sixth row of the table.
(There is no number xsuch that 6 ×x= 1.) That not
every fraction is represented in clock math is deemed a
deficiency of the system.
Generalized Clock Math: Modular Arithmetic
One can envision a clock with a different number of
hours represented on its face. For example, in 5 o’clock
math, just five hours are depicted, 0, 1, 2, 3, and 4
(again it is appropriate to deem the fifth hour as the
same as the zeroth hour), and all other numbers are
replaced by their excess over a multiple of 5. Thus, for
example, we have 6 ≡1 (mod 5) and 32 ≡2 (mod 5).
The product table for mod-5 arithmetic appears as fol-
lows. Notice that every digit appears in every (nonzero)
row of the table.
In general, in base Nmodular arithmetic, each number
is replaced by its excess over a multiple of N.
If Nis a
COMPOSITE NUMBER
, Nequals a×bsay,
then the entry in the ath row and bth column of the
product table is zero (for example, 3 ×4 is zero in
12-clock math). Consequently the number zero
appears in the ath row more than once, giving insuffi-
cient space for all the other digits to appear in that
row. On the other hand, if Nis
PRIME
, then each and
every nonzero row of the product table does indeed
contain every digit, as demonstrated by the product
table for 5-clock math. (To see why this is the case,
note that for any number aless than the prime Nis
RELATIVELY PRIME
to N, and so, by the E
UCLIDEAN
ALGORITHM
, there exist integers xand ysuch that ax
+ yN = 1. This shows that ax is 1 more than a multi-
ple of N, and so ax ≡1(mod N). Consequently the
number 1 appears in the ath row, xth column, of the
product table. So too does the number 2, since a(2x)
= 2ax ≡2 (mod N), the number 3, a(3x) = 3ax ≡3
(mod N), and so forth.) We have:
If Nis a prime number, then all digits appear
in each nonzero row of the product table for
arithmetic modulo N.
In particular, for any nonzero number ain mod-N
arithmetic, the number 1 appears in the ath row of
the product table. Thus there is a number xsuch that
a×x= 1. The fraction thus has a valid interpretation
in this system. This argument applies to all fractions one
may wish to consider. We have:
If Nis a prime number, then all fractions exist
in mod-Narithmetic.
This completely classifies all modular arithmetic sys-
tems that possess fractions.
1
–
a
modular arithmetic 341
x 0 1 2 3 4 5 6 7 8 9 10 11
00000000000 00
1 0 1 2 3 4 5 6 7 8 9 10 11
202468100246 810
30369036903 69
40480480480 48
50 51038 1 6114 9 2 7
60606060606 06
707294116183105
80840840840 84
90963096309 63
10 0 10 8 6 4 2 0 10 8 6 4 2
11011109876543 21
x01234
000000
101234
202413
303142
404321