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—the modulus. 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 is divided 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 the answer 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 9:00 the next day, rather than 33:00. Since the hour number starts over after it reaches 12, this is arithmetic modulo 12.^[1]

1 Congruence relation
2 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

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 positive integer n, two integers a and b are said to be congruent modulo n, written:

$a \equiv b \pmod n,\,$

if their difference a − b is an integer multiple of n. The number n is called the modulus of the congruence. An equivalent definition is that both numbers have the same remainder when divided by n.

For example,

$38 \equiv 14 \pmod {12}\,$

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,

$38 \equiv 2 \pmod {12}\,$

because both numbers, when divided by 12, have the same remainder (2). Equivalently, the fractional parts of doing a full division of each of the numbers by 12 are the same: 0.1666... (38/12 = 3.1666..., 2/12 = 0.1666...). From the prior definition we also see that their difference, a − b = 36, is a whole number (integer) multiple of 12 (n = 12, 36/12 = 3).

The same rule holds for negative values of a:

$-3 \equiv 2 \pmod 5.\,$

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.

$a_1 \equiv b_1 \pmod n$

and

$a_2 \equiv b_2 \pmod n,$

then:

$a_1 + a_2 \equiv b_1 + b_2 \pmod n\,$
$a_1 - a_2 \equiv b_1 - b_2 \pmod n\,$

It should be noted that the above two properties would still hold if the theory were expanded to include all real numbers, that is if $a_1, a_2, b_1, b_2, n\,$ were not necessarily all integers. The next property, however, would fail if these variables were not all integers:

$a_1 a_2 \equiv b_1 b_2 \pmod n.\,$

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 $\overline{a}_n$ , is the set $\left\{\ldots, a - 2n, a - n, a, a + n, a + 2n, \ldots \right\}$ . 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 $\displaystyle [a]$ .

The set of congruence classes modulo n is denoted as $\mathbb{Z}/n\mathbb{Z}$ (or, alternatively, $\mathbb{Z}/n$ or $\mathbb{Z}_n$ ) and defined by:

$\mathbb{Z}/n\mathbb{Z} = \left\{ \overline{a}_n | a \in \mathbb{Z}\right\}.$

When n ≠ 0, $\mathbb{Z}/n\mathbb{Z}$ has n elements, and can be written as:

$\mathbb{Z}/n\mathbb{Z} = \left\{ \overline{0}_n, \overline{1}_n, \overline{2}_n,\ldots, \overline{n-1}_n \right\}.$

When n = 0, $\mathbb{Z}/n\mathbb{Z}$ does not have zero elements; rather, it is isomorphic to $\mathbb{Z}$ , since $\overline{a}_0 = \left\{a\right\}$ .

We can define addition, subtraction, and multiplication on $\mathbb{Z}/n\mathbb{Z}$ by the following rules:

$\overline{a}_n + \overline{b}_n = \overline{(a + b)}_n$
$\overline{a}_n - \overline{b}_n = \overline{(a - b)}_n$
$\overline{a}_n \overline{b}_n = \overline{(ab)}_n.$

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

In this way, $\mathbb{Z}/n\mathbb{Z}$ becomes a commutative ring. For example, in the ring $\mathbb{Z}/24\mathbb{Z}$ , we have

$\overline{12}_{24} + \overline{21}_{24} = \overline{9}_{24}$

as in the arithmetic for the 24-hour clock.

The notation $\mathbb{Z}/n\mathbb{Z}$ is used, because it is the factor ring of $\mathbb{Z}$ by the ideal $n\mathbb{Z}$ containing all integers divisible by n, where $0\mathbb{Z}$ is the singleton set $\left\{0\right\}$ . Thus $\mathbb{Z}/n\mathbb{Z}$ is a field when $n\mathbb{Z}$ is a maximal ideal, that is, when $n$ is prime.

In terms of groups, the residue class $\overline{a}_n$ is the coset of a in the quotient group $\mathbb{Z}/n\mathbb{Z}$ , a cyclic group.

The set $\mathbb{Z}/n\mathbb{Z}$ 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 $\mathbb{Z}/0\mathbb{Z}$ (which, as mentioned before, is isomorphic to the ring $\mathbb{Z}$ of integers), for example when discussing the characteristic of a ring.

Remainders

The notion of modular arithmetic is related to that of the remainder in division. The operation of 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 equality indicated by "=". Equality implies specifically the "common residue", the least non-negative member of an equivalence class. When working with modular arithmetic, each equivalence class is usually represented by its common residue, for example 38 ≡ 2 (mod 12) which can 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 are negative. Hence to express the remainder we would have to write −5 ≡ −17 (mod 12), rather than 7 = −17 (mod 12), since equivalence can only be said of common residues with the same sign.

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 BASIC, SQL, Haskell), with exceptions (e.g. Excel). These operators are commonly pronounced as "mod", but it is specifically a remainder that is computed (since in C99 negative number will be returned if the first argument is negative, and in Python a negative number will be returned if the second argument is negative). The function modulo instead of mod, like 38 ≡ 14 (modulo 12) is sometimes used to 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 mod 12, 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.

Functional representation of the remainder

Computation with modular arithmetic can be implemented using other functions.

One such functional representation uses the floor function. If a ≡ b (mod n), a ≥ 0, and 0 ≤ b < n, then there exists an integer k ≥ 0 such that a = kn + b. The remainder b can be written

$b = a - \left\lfloor \frac{a}{n} \right\rfloor \times n,$

where $\left\lfloor \frac{a}{n} \right\rfloor \,$ is the largest integer less than or equal to $\frac{a}{n}$ . If instead −n ≤ b < 0, then

$b = a - \left\lfloor \frac{a}{n} \right\rfloor \times n - n.$

Another functional representation uses sine and arcsine, taking advantage of the fact that arcsine is multivalued. Let

$g(x) = \frac{2n}{\pi} \arcsin \left( \sin \left( \frac{\pi x}{2n} \right)\right).$

Then

g(x) = x (mod n)

for 0 ≤ πx/2n < π/2 or π ≤ πx/2n < 3π/2, and

g(x) = −x (mod n)

for π/2 ≤ πx/2n < π or 3π/2 ≤ πx/2n < 2π.

Applications

Modular arithmetic is referenced in number theory, group theory, ring theory, knot theory, abstract algebra, cryptography, computer science, chemistry 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.

Modular arithemetic is often used to calculate checksums that are used within identifiers - International Bank Account Numbers (IBANs) for example make use of modulo 97 arithmetic to trap user input errors in bank account numbers.

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. XOR is the modular addition of two bits.

In chemistry, the last digit of the CAS registry number (a number which is unique for each chemical compound) is a check digit, which is calculated by taking the last digit of the first two parts of the CAS registry number times 1, the next digit times 2, the next digit times 3 etc., adding all these up and computing the sum modulo 10.

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.

Computational complexity

Since modular arithmetic has such a wide range of applications, it is important to know how hard it is to solve a system of congruences. A linear system of congruences can be solved in polynomial time with a form of Gaussian elimination, for details see linear congruence theorem. Algorithms, such as Montgomery reduction, also exist to allow simple arithmetic operations, such as multiplication and exponentiation modulo n, to be performed efficiently on large numbers.

Solving a system of non-linear modular arithmetic equations is NP-complete.^[2]

Notes

↑ Technically, if the time were represented modulo 12, then 12:00 would have to be "0:00" because $12 \equiv 0 \pmod {12}$ . Perhaps a better example would be 24 hour time, which uses 0:00 for midnight. 24 time would be modulo 24.
↑ M. R. Garey, D. S. Johnson: Computers and Intractability, a Guide to the Theory of NP-Completeness, W. H. Freeman 1979.

References

Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, MR 0434929, ISBN 978-0-387-90163-3 . See in particular chapters 5 and 6 for a review of basic modular arithmetic.
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.
Anthony Gioia, Number Theory, an Introduction Reprint (2001) Dover. ISBN 0-486-41449-3

External links

In this modular art article, one can learn more about applications of modular arithmetic in art.
Weisstein, Eric W., "Modular Arithmetic" from MathWorld.
An article on modular arithmetic on the GIMPS wiki
Modular Arithmetic and patterns in addition and multiplication tables
Whitney Music Box - an audio/video demonstration of integer modular math
Automated modular arithmetic theorem provers:
- Spear
- AAProver - Simple C++ framework easy to use in other applications, supporting (among others) all integer operators present in languages such as C/C++/Java and arbitrary bit-width.