Basic Example
The prototypical example of a congruence relation is congruence modulo on the set of integers. For a given positive integer, two integers and are called congruent modulo , written
if is divisible by (or equivalently if and have the same remainder when divided by ).
for example, and are congruent modulo ,
since is a multiple of 10, or equivalently since both and have a remainder of when divided by .
Congruence modulo (for a fixed ) is compatible with both addition and multiplication on the integers. That is, if
- and
then
- and
The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo is a congruence relation on the ring of integers, and arithmetic modulo occurs on the corresponding quotient ring.
Read more about this topic: Congruence Relation
Famous quotes containing the word basic:
“The man who is admired for the ingenuity of his larceny is almost always rediscovering some earlier form of fraud. The basic forms are all known, have all been practicised. The manners of capitalism improve. The morals may not.”
—John Kenneth Galbraith (b. 1908)
“Not many appreciate the ultimate power and potential usefulness of basic knowledge accumulated by obscure, unseen investigators who, in a lifetime of intensive study, may never see any practical use for their findings but who go on seeking answers to the unknown without thought of financial or practical gain.”
—Eugenie Clark (b. 1922)