Here we have two different types of arithmetic system, on one
hand 'modulus' (or clock) arithmetic and on the other hand, place
value systems (which use different number bases). Unlike modulus
arithmetic, where many numbers are represented by the same symbol,
in base arithmetic each distinct number has its own distinct
representation different from that of any other number.
The table compares counting in the two systems, using arithmetic
modulo 7 and numbers in base 7 as an example. The units digit of a
number in base 7 is the same as the equivalent number in arithmetic
modulo 7.
| Clock arith. |
1 |
2 |
3 |
4 |
5 |
6 |
0 |
1 |
2 |
3 |
4 |
5 |
6 |
0 |
1 |
2 |
3 |
| Base arith. |
1 |
2 |
3 |
4 |
5 |
6 |
10 |
11 |
12 |
13 |
14 |
15 |
16 |
20 |
21 |
22 |
23 |
In base 7, numbers have the units digits 0, 1, 2, 3, 4, 5 and
6. In arithmetic modulo 7, any integer $m$ is represented by (or
said to be congruent to) one of the numbers 0, 1, 2, 3, 4, 5, or 6
which is given by the remainder when $m$ is divided by 7.
For example the decimal number 33 is written as 45 in the base
7 because 33 = 4$\times$7 $+$ 5 and 33 is congruent to the number 5
in arithmetic modulo 7. The decimal number 80 is written 143 in the
base 7 system because 80 = 11$\times$7 $+$ 3 = 1$\times$49 $+$
4$\times$7 $+$ 3 and 80 is congruent to the number 3 in arithmetic
modulo 7.
Vassil Vassilev, age 15, Lawnswood High School, Leeds gave the
following proof that, in modulus arithmetic, when we add or
multiply two numbers, we get the same result as when we first find
numbers to which they are congruent and then add or multiply those
numbers.
If $m = p + 7r$ and $n = q + 7s$ (where $m, n, p, q, r, s$ are
integers) we can say that $m$ and $n$ are congruent to $p$ and $q$
modulo 7 respectively and we write: $m \equiv p {\rm mod} 7$ and $n
\equiv q {\rm mod} 7$.
$m + n = p + 7r + q + 7s$
$m + n = p + q + 7(r + s)$, where
$r + s$ is an integer.
$m + n \equiv (p + q) {\rm mod \ }
7$.
$mn = (p + 7r)(q + 7s)$
$mn = pq + 7ps +7rq +49rs$
$= pq + 7(ps + rq + 7rs)$, where
$ps + rq + 7rs$ is an integer.
$mn \equiv (pq) {\rm mod \ }
7$.