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MODULAR ARITHMETIC
In mathematics , modular arithmetic (sometimes called clock
arithmetic ) is a system of arithmetic for integers , where numbers
"wrap around" after they reach a certain value themodulus . Modular
arithmetic was introduced by Carl Friedrich Gauss in his book
Disquisitiones Arithmeticae , published in 1801.
Time-keeping on a clock gives an example of modular
arithmetic.
A familiar use of modular arithmetic is its use in the 12-hour
clock , in which the day isdivided into two 12 hour periods. If the
time is 7:00 now, then 8 hours later it will be 3:00.Usual addition
would suggest that the later time should be 7 + 8 = 15, but this is
not theanswer because clock time "wraps around" every 12 hours;
there is no "15 o'clock".Likewise, if the clock starts at 12:00
(noon) and 21 hours elapse, then the time will be 9:00the next day,
rather than 33:00. Since the hour number starts over when it
reaches 12, this isarithmetic modulo 12.[1]
Contents
[hide ]
1 The congruence relation 2 The ring of congruence classes 3
Remainders 4 Functional representation of the remainder 5
Applications 6 Computational complexity 7 See also 8 Notes 9
References 10 External links
[edit ] The congruence relationModular arithmetic can be handled
mathematically by introducing a congruence relation onthe integers
that is compatible with the operations of the ring of integers:
addition , subtraction , and multiplication . For a fixed modulus
n, it is defined as follows.
Two integers a and b are said to be congruent modulo n, if their
difference a b is aninteger multiple of n. An equivalent definition
is that both numbers have the same remainder when divided by n. If
this is the case, it is expressed as:
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MODULAR ARITHMETIC
When n 0, has n elements, and can be written as:
When n = 0, does not have zero elements; rather, it is
isomorphic to , since.
We can define addition, subtraction, and multiplication on by
the following rules:
The verification that this is a proper definition uses the
properties given before.
In this way, becomes a commutative ring . For example, in the
ring , we have
as in the arithmetic for the 24-hour clock.
The notation is used, because it is the factor ring of by the
ideal containing allintegers divisible by n, where is the singleton
set . Thus is a field whenis a maximal ideal , that is, when n is
prime.
In terms of groups, the residue class is the coset of a in the
quotient group , acyclic group .
The set has a number of important mathematical properties that
are foundational tovarious branches of mathematics.
Rather than excluding the special case n = 0, it is more useful
to include (which, asmentioned before, is isomorphic to the ring of
integers), for example when discussing thecharacteristic of a ring
.
[edit ] Remainders
The notion of modular arithmetic is related to that of the
remainder in division . The operationof finding the remainder is
sometimes referred to as the modulo operation and we may see 2= 14
(mod 12). The difference is in the use of congruency, indicated by
, and equalityindicated by =. Equality implies specifically the
"common residue", the least non-negativemember of an equivalence
class. When working with modular arithmetic, each equivalenceclass
is usually represented by its common residue, for example 3 8 2
(mod 12) which can
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MODULAR ARITHMETIC
be found using long division . It follows that, while it is
correct to say 38 14 (mod 12), and 2 14 (mod 12), it is incorrect
to say 38 = 14 (mod 12) (with "=" rather than "").
The difference is clearest when dividing a negative number,
since in that case remainders arenegative. Hence to express the
remainder we would have to write 5 17 (mod 12), rather
than 7 = 17 (mod 12), since "=" can only be used with the
(positive) common residue.
In computer science , it is the remainder operator that is
usually indicated by either "%" (e.g.in C, Java , Javascript , and
Python ) or "mod" (e.g. in SQL , Visual Basic ), with
exceptions(e.g. Excel). These operators are commonly pronounced as
"mod", but it is specifically aremainder that is computed (since a
negative number will be returned if the first argument isnegative).
The function modulo instead of mod , like 38 14 (modulo 12) is
sometimes usedto indicate the common residue rather than a
remainder (e.g. in Ruby ).
Parentheses are sometimes dropped from the expression, e.g. 38
14 mod 12 or 2 = 14 mod12, or placed around the divisor e.g. 38 14
mod (12). Notation such as 38(mod 12) has also
been observed, but is ambiguous without contextual
clarification
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