Modular arithmetic

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Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic because of its use in the 24-hour clock system) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus. Modular arithmetic was introduced by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801.

A familiar use of modular arithmetic is its use in the 24-hour clock: the arithmetic of time-keeping in which the day runs from midnight to midnight and is divided into 24 hours, numbered from 0 to 23. If the time is noted at 7 o'clock in the evening — 19:00 in the 24-hour system — and then again 8 hours later, then rather than the time being 27:00 (as in usual addition: 19 + 8 = 27), the time will actually be denoted as 03:00, albeit in the next day. Likewise, if the clock starts at noon (12:00) and 21 hours elapse, then the time will be 09:00 the next day, rather than 33:00 (as in usual addition). Since the hour number starts over at 00 hours after passing 23 hours, this is arithmetic modulo 24 — the hours "wrap around" upon reaching the modulus 24.

1 The congruence relation
2 The ring of congruence classes
3 Remainders
4 Applications
5 References
6 See also
7 External links

[edit] The congruence relation

Modular arithmetic can be handled mathematically by introducing a congruence relation on the 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 an integer multiple of n. If this is the case, it is expressed as:

$a \equiv b \text{ (mod } n)$

The above mathematical statement is read: "a is congruent to b modulo n".

For example,

$38 \equiv 14 \text{ (mod } 12)$

because 38 − 14 = 24, which is a multiple of 12. For positive n and non-negative a and b, congruence of a and b can also be thought of as asserting that these two numbers have the same remainder after dividing by the modulus n. So,

$38 \equiv 14 \text{ (mod } 12)$

because, when divided by 12, both numbers give 2 as remainder.

A remark on the notation: Because it is common to consider several congruence relations for different moduli at the same time, the modulus is incorporated in the notation. In spite of the ternary notation, the congruence relation for a given modulus is binary. This would have been clearer if the notation a ≡_n b had been used, instead of the common traditional notation.

The properties that make this relation a congruence relation (respecting addition, subtraction, and multiplication) are the following.

If $a_{1} \equiv b_{1} \text{ (mod } n)$ and $a_{2} \equiv b_{2} \text{ (mod } n)$ , then:

$(a_{1} + a_{2}) \equiv (b_{1} + b_{2}) \text{ (mod } n)$
$(a_{1} - a_{2}) \equiv (b_{1} - b_{2}) \text{ (mod } n)$
$(a_{1} a_{2}) \equiv (b_{1} b_{2}) \text{ (mod } n)$

[edit] The ring of congruence classes

Like any congruence relation, congruence modulo n is an equivalence relation, and the equivalence class of the integer a, denoted by [a]_n, is the set { ..., a − 2n, a − n, a, a + n, a + 2n, ...}. This set, consisting of the integers congruent to a modulo n, is called the congruence class or residue class of a modulo n. Another notation for this congruence class, which requires that in the context the modulus is known, is $\hat{a}$ .

The set of congruence classes modulo n is denoted as Z/nZ and defined by:

Z/nZ = { [a]_n | a ∈ Z}, where Z = {..., −2, −1, 0, 1, 2, ...}.

When n ≠ 0, Z/nZ has |n| elements, and can be written as:

Z/nZ = { [0]_n, [1]_n, [2]_n, ..., [|n|−1]_n }

When n = 0, Z/nZ does not have zero elements; rather, it is isomorphic to Z, since [a]₀ = {a}.

We can define addition, subtraction, and multiplication on Z/nZ by the following rules:

[a]_n + [b]_n = [a + b]_n
[a]_n − [b]_n = [a − b]_n
[[a]_n × [b]_n]_n = [ab]_n

The verification that this is a proper definition uses the properties given before.

In this way, Z/nZ becomes a commutative ring. For example, in the ring Z/24Z, we have

[12]₂₄ + [21]₂₄ = [9]₂₄,

as in the arithmetic for the 24-hour clock.

The notation Z/nZ is used, because it is the factor ring of Z by the ideal nZ containing all integers divisible by n, where 0Z is the singleton set {0}.

In terms of groups, the residue class [a]_n is the coset of a in the quotient group Z/nZ, a cyclic group.

The set Z/nZ has a number of important mathematical properties that are foundational to various branches of mathematics. Rather than excluding the special case n = 0, it is more useful to include Z/0Z (which, as mentioned before, is isomorphic to the ring Z of integers), for example when discussing the characteristic of a ring.

[edit] Remainders

The notion of modular arithmetic is related to that of the remainder in division. The operation of finding the remainder is known as the modulo operation and is sometimes written as "mod", so we write "14 mod 12 = 2". This meaning of "mod" is subtly but significantly different from that introduced in this article; it is true to say "38 ≡ 14 (mod 12)" , but it is not true to say "38 = 14 mod 12" — 38 is congruent to 14 modulo 12, but the remainder of 14 divided by 12 is 2, not 38. To avoid this confusion, the congruence relation is sometimes expressed by using modulo instead of mod like "38 ≡ 14 (modulo 12)" in computer science.

When working with modular arithmetic, each equivalence class is usually represented with its least non-negative member, which is called the common residue. This can be found using long division.

[edit] Applications

Modular arithmetic is referenced in number theory, group theory, ring theory, abstract algebra, cryptography, computer science, and the visual and musical arts.

It is one of the foundations of number theory, touching on almost every aspect of its study, and provides key examples for group theory, ring theory and abstract algebra.

In cryptography, modular arithmetic directly underpins public key systems such as RSA and Diffie-Hellman, as well as providing finite fields which underlie elliptic curves, and is used in a variety of symmetric key algorithms including AES, IDEA, and RC4.

In computer science, modular arithmetic is often applied in bitwise operations and other operations involving fixed-width, cyclic data structures. The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context.

In the visual arts, modular arithmetic can be used to create artistic patterns based on the multiplication and addition tables modulo n (see external link, below).

In music, arithmetic modulo 12 is used in the consideration of the system of twelve-tone equal temperament, where octave and enharmonic equivalency occurs (that is, pitches in a 1∶2 or 2∶1 ratio are equivalent, and C-sharp is considered the same as D-flat).

The method of casting out nines offers a quick check of decimal arithmetic computations performed by hand. It is based on modular arithmetic modulo 9, and specifically on the crucial property that 10 ≡ 1 (mod 9).

More generally, modular arithmetic also has application in disciplines such as law (see e.g., apportionment), economics, (see e.g., game theory) and other areas of the social sciences, where proportional division and allocation of resources plays a central part of the analysis.

Some neurologists (see e.g., Oliver Sacks) theorize that so-called autistic savants utilize an "innate" modular arithmetic to compute such complex problems as what day of the week a distant date will fall on.

[edit] References

Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. See in particular chapters 5 and 6 for a review of basic modular arithmetic. ISBN 0-387-90163-9
Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. Introduction to Algorithms, Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Section 31.3: Modular arithmetic, pp.862–868.