MORE ABOUT
NUMBERS
What is congruence?
In reference to numbers, congruence is
the property of two integers having the
same remainder upon division by another
integer. The term also is often used in
geometry to describe a property of geo-
metric formations (for more information
about congruence in geometry, see
“Geometry and Trigonometry”). Still
another way of using congruence is in
number theory, in which modular arith-
metic is the arithmetic of congruences,
which is sometimes informally called
“clock arithmetic.”How does modular arithmeticwork?
In modular arithmetic, numbers “wrap around” when they reach a fixed quantity. This
is also called the modulus—thus the name modular arithmetic—with the standard
way of writing the form as “mod 12” or “mod 2.”In this case, if the two numbers b(also called the base) and c(also called the
remainder) are subtracted (bc), and their difference is a number integrally divisible
by m,or (bc)/m,then band care said to be congruent modulo m. Mathematically,
“bis congruent to c(modulo m)” is written as follows, with the symbol for congru-
ence ( ):b c(mod m)But if bcis not integrally divisible by m, then it is said, “bis not congruent to c
(modulo m),” orb c(mod m)More formally, modular arithmetic includes any “non-trivial homomorphic image
of the ring of integers.” We can interpret this interesting definition using a clock. The
modulus would be the number 12 on the clock (arithmetic modulo 12), with an asso-
ciated ring labeled C 12 and the allowable numbers being 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
and 11. Another example is arithmetic modulo 2, with an associated ring of C 2 , or
82 allowable numbers of 1 and 2.
We can look to the familiar face of a clock to illus-
trate the concept of “clock arithmetic” or modulo- With clocks, we can count through 11 before
getting to 12, where we start over again at zero.
Thus, if it is 7 o’clock and we add 6 hours, we get 1
o’clock, not 13 o’clock. Taxi/Getty Images.