Modular arithmetic - Citizendia

Modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value — the modulus. Arithmetic or arithmetics (from the Greek word αριθμός = number is the oldest and most elementary branch of mathematics used by almost everyone The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French Modular arithmetic was introduced by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801. Johann Carl Friedrich Gauss (ˈɡaʊs, Gauß Carolus Fridericus Gauss ( 30 April 1777 – 23 February 1855) was a German The Disquisitiones Arithmeticae is a textbook of Number theory written by German Mathematician Carl Friedrich Gauss in 1798

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. Description A time of day is written in the 24-hour notation in the form hhmm (for example 0123 or hhmmss (for example 012345 where hh (00 to 23 is the decimal number If the time is 19:00 now — 7 o'clock in the evening — then 8 hours later it will be 3:00. Usual addition would suggest that the later time should be 19 + 8 = 27, but this is not the answer because clock time "wraps around" at the end of the day. Likewise, if the 24-hour clock starts at 12:00 (noon) and 21 hours elapse, then the time will be 09:00 the next day, rather than 33:00. Since the hour number starts over when it reaches 24, this is arithmetic modulo 24. Note: The clock shown below is not a 24-hour clock, it's the more widely used 12-hour, "modulo" 12, clock.

Time-keeping on a clock gives an example of modular arithmetic.

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

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. See Congruence (geometry for the term as used in elementary geometry The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real Addition is the mathematical process of putting things together Subtraction is one of the four basic Arithmetic operations it is the inverse of Addition, meaning that if we start with any number and add any number and then subtract 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. In Mathematics, a multiple of an Integer is the product of that integer with another integer If this is the case, it is expressed as:

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

The above mathematical statement is read: "a is congruent to b modulo 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. In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left So,

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

because, when divided by 12, both numbers have the same remainder, . 1666. . . (38/12 = 3. 166. . . , 2/12 = . 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 integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French

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. In Mathematics, a binary relation (or a dyadic or 2-place relation) is an arbitrary association of elements within a set or with elements of 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 $a_1 \equiv b_1 \pmod n$ and $a_2 \equiv b_2 \pmod n$ , then:

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

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 $\overline{a}_n$ , is the set $\left\{\ldots, a - 2n, a - n, a, a + n, a + 2n, \ldots \right\}$ . In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X 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 $\mathbb{Z}/n\mathbb{Z}$ 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 $\overline{a}_0 = \left\{a\right\}$ . In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

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. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property 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 $\left\{0\right\}$ . In Mathematics a quotient ring, also known as factor ring or residue class ring, is a construction in Ring theory, quite similar to the In Ring theory, a branch of Abstract algebra, an ideal is a special Subset of a ring. In Mathematics, a singleton is a set with exactly one element

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. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an

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. In Mathematics, the characteristic of a ring R, often denoted char( R) is defined to be the smallest number of times one must add the ring's In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real

Remainders

The notion of modular arithmetic is related to that of the remainder in division. In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. 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. Long Division is the second album by the Rustic Overtones, originally released on November 17 1995 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 "≡").

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.

The congruence relation is sometimes expressed by using modulo instead of mod, like "38 ≡ 14 (modulo 12)" in computer science. Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their The modulo function in various computer languages typically yield the common residue, for example the statement "y = MOD(38,12);" gives y = 2.

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. Number theory is the branch of Pure mathematics concerned with the properties of Numbers in general and Integers in particular as well as the wider classes Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, ring theory is the study of rings, Algebraic structures in which addition and multiplication are defined and have similar properties to those In Mathematics, knot theory is the area of Topology that studies mathematical knots While inspired by knots which appear in daily life in shoelaces Abstract algebra is the subject area of Mathematics that studies Algebraic structures such as groups, rings, fields, modules Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" Computer science (or computing science) is the study and the Science of the theoretical foundations of Information and Computation and their Chemistry (from Egyptian kēme (chem meaning "earth") is the Science concerned with the composition structure and properties The visual arts are art forms that focus on the creation of works which are primarily Visual in nature such as Painting, Photography Music is an Art form in which the medium is Sound organized in Time.

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. Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key In Cryptography, RSA is an Algorithm for Public-key cryptography. Diffie-Hellman key exchange ( D-H) is a Cryptographic protocol that allows two parties that have no prior knowledge of each other to jointly establish a shared secret In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements In Mathematics, an elliptic curve is a smooth, projective Algebraic curve of genus one on which there is a specified point O Symmetric-key algorithms are a class of Algorithms for Cryptography that use trivially related often identical Cryptographic keys for both decryption In Cryptography, the Advanced Encryption Standard ( AES) also known as Rijndael, is a Block cipher adopted as an Encryption In Cryptography, the International Data Encryption Algorithm ( IDEA) is a Block cipher designed by Xuejia Lai and James Massey In Cryptography, RC4 (also known as ARC4 or ARCFOUR meaning Alleged RC4 see below is the most widely-used software

In computer science, modular arithmetic is often applied in bitwise operations and other operations involving fixed-width, cyclic data structures. In Computer programming, a bitwise operation operates on one or two Bit patterns or binary numerals at the level of their individual Bits On most A data structure in Computer science is a way of storing Data in a computer so that it can be used efficiently The modulo operation, as implemented in many programming languages and calculators, is an application of modular arithmetic that is often used in this context. A programming language is an Artificial language that can be used to write programs which control the behavior of a machine particularly a Computer. A calculator is device for performing mathematical calculations distinguished from a Computer by having a limited problem solving ability and an interface optimized for interactive

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. CAS registry numbers are unique numerical identifiers for Chemical compounds Polymers biological sequences mixtures and Alloys They are also referred to A check digit is a form of Redundancy check used for Error detection, the decimal equivalent of a binary Checksum. CAS registry numbers are unique numerical identifiers for Chemical compounds Polymers biological sequences mixtures and Alloys They are also referred to , adding all these up and computing the sum modulo 10.

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). Equal temperament is a Musical temperament, or a system of tuning in which every pair of adjacent notes has an identical Frequency ratio. In Music, an octave ( is the the use of which is "common in most musical systems In modern Music and notation, an enharmonic equivalent is a Note ( enharmonic tone) interval ( enharmonic interval) or In Music, sharp means higher in pitch More specifically in Musical notation, sharp means "higher in pitch by a Semitone (half step" In Music, flat means "lower in pitch" More specifically in Music notation, flat means "lower in pitch by a Semitone

The method of casting out nines offers a quick check of decimal arithmetic computations performed by hand. Casting out nines is a Sanity check to ensure that hand computations of sums differences products and quotients of Integers are correct 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. Law is a system of rules enforced through a set of Institutions used as an instrument to underpin civil obedience politics economics and society g. , apportionment), economics, (see e. Economics is the social science that studies the production distribution, and consumption of goods and services. g. , game theory) and other areas of the social sciences, where proportional division and allocation of resources plays a central part of the analysis. Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering, The social sciences comprise academic disciplines concerned with the study of the social life of human groups and individuals including Anthropology, Communication studies Proportional division or simple fair division is the original and simplest problem in Fair division.

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. Oliver Wolf Sacks, CBE (born July 9, 1933, London is a British Neurologist residing in the United States who has written popular books about Savant syndrome —sometimes abbreviated as savantism —is not a recognized medical diagnosis but researcher Darold Treffert defines it as a rare condition in which persons with

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 the linear congruence theorem. In Linear algebra, Gaussian elimination is an efficient Algorithm for solving systems of linear equations, to find the rank of a matrix In Modular arithmetic, the question of when a linear congruence can be solved is answered by the linear congruence theorem.

Solving a system of non-linear modular arithmetic equations is NP-complete. In Computational complexity theory, the Complexity class NP-complete (abbreviated NP-C or NPC) is a class of problems having two properties For details, see for example M. R. Garey, D. S. Johnson: Computers and Intractability, a Guide to the Theory of NP-Completeness, W. H. Freeman 1979.

References

Tom M. Apostol, Introduction to Analytic Number Theory, (1976) Springer-Verlag, New York. The Chinese remainder theorem is a result about congruences in Number theory and its generalizations in Abstract algebra. Lagrange's theorem, in the Mathematics of Group theory, states that for any Finite group G, the order (number of elements of The word modulo (Latin with respect to a modulus of ___ is the Latin Ablative of Modulus which itself means "a small measure In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Arithmetic, when the result of the division of two Integers cannot be expressed with an integer Quotient, the remainder is the amount "left In Mathematics, the n th Pisano period, written π( n) is the period with which the Sequence of Fibonacci numbers modulo In Mathematics, a Boolean ring R is a ring (with identity for which x 2 = x for all x in R; that Tom Mike Apostol (born 1923 is an American analytic number theorist and professor at the California Institute of Technology. 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. Thomas H Cormen is the co-author of Introduction to Algorithms, along with Charles Leiserson, Ron Rivest, and Cliff Stein. Charles Eric Leiserson is a Computer scientist, specializing in the theory of Parallel computing and Distributed computing, and particularly practical Ronald Linn Rivest (born 1947, Schenectady, New York) is a cryptographer. Clifford Stein, a Computer scientist, is currently a professor of Industrial engineering and Operations research at Columbia University Introduction to Algorithms, Second Edition. Introduction to Algorithms is a book by Thomas H Cormen, Charles E 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.
Eric W. Weisstein, Modular Arithmetic at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
An article on modular arithmetic on the GIMPS wiki
Modular Arithmetic and patterns in addition and multiplication tables
Automated modular arithmetic theorem provers:
- BAT
- Spear
- STP

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