modular arithmetic
arithmetic in which numbers that are congruent modulo a given number are treated as the same. Cf. congruence (def. 2), modulo, modulus (def. 2b).
[1955-60]

* * *

sometimes referred to as  modulus arithmetic  or  clock arithmetic 

      in its most elementary form, arithmetic done with a count that resets itself to zero every time a certain whole number N greater than one, known as the modulus (mod), has been reached. Examples are a digital clock in the 24-hour system, which resets itself to 0 at midnight (N = 24), and a circular protractor marked in 360 degrees (N = 360). Modular arithmetic is important in number theory, where it is a fundamental tool in the solution of Diophantine equations (Diophantine equation) (particularly those restricted to integer solutions). Generalizations of the subject led to important 19th-century attempts to prove Fermat's last theorem and the development of significant parts of modern algebra (algebra, modern).

      Under modular arithmetic (with mod N), the only numbers are 0, 1, 2, …, N − 1, and they are known as residues modulo N. Residues are added by taking the usual arithmetic sum, then subtracting the modulus from the sum as many times as is necessary to reduce the sum to a number M between 0 and N − 1 inclusive. M is called the sum of the numbers modulo N. Using notation introduced by the German mathematician Carl Friedrich Gauss (Gauss, Carl Friedrich) in 1801, one writes, for example, 2 + 4 + 3 + 7 ≡ 6 (mod 10), where the symbol ≡ is read “is congruent to.”

      Examples of the use of modular arithmetic occur in ancient Chinese, Indian, and Islamic cultures. In particular, they occur in calendrical and astronomical problems since these involve cycles (man-made or natural), but one also finds modular arithmetic in purely mathematical problems. An example from a 3rd-century-AD Chinese book, Sun Zi's Sunzi suanjing (Master Sun's Mathematical Manual), asks

We have a number of things, but we do not know exactly how many. If we count them by threes we have two left over. If we count by fives we have three left over. If we count by sevens there are two left over. How many things are there?

      This is equivalent to asking for the solution of the simultaneous congruences X ≡ 2 (mod 3), X ≡ 3 (mod 5), and X ≡ 2 (mod 7), one solution of which is 23. The general solution of such problems came to be known as the Chinese remainder theorem.

      The Swiss mathematician Leonhard Euler (Euler, Leonhard) pioneered the modern approach to congruence about 1750, when he explicitly introduced the idea of congruence modulo a number N and showed that this concept partitions the integers into N congruence classes, or residue classes. Two integers are in the same congruence class modulo N if their difference is divisible by N. For example, if N is 5, then −6 and 4 are members of the same congruence class {…, −6, −1, 4, 9, …}. Since each congruence class may be represented by any of its members, this particular class may be called, for example, “the congruence class of −6 modulo 5” or “the congruence class of 4 modulo 5.”

      In Euler's system any N numbers that leave different remainders on division by N may represent the congruence classes modulo N. Thus, one possible system for arithmetic modulo 5 would be −2, −1, 0, 1, 2. Addition of congruence classes modulo N is defined by choosing any element from each class, adding the elements together, and then taking the congruence class modulo N that the sum belongs to as the answer. Euler similarly defined subtraction and multiplication of residue classes. For example, to multiply −3 by 4 (mod 5), first multiply −3 × 4 = −12; since −12 ≡ 3 (mod 5), the solution is −3 × 4 ≡ 3 (mod 5). Euler showed that one would get the same result with any two elements from the corresponding congruence classes.

      Note that when the modulus N is not prime, division is not always possible. For example, 1 ÷ 2 ≡ 3 (mod 5), since 2 × 3 ≡ 1 (mod 5). However, the equation 1 ÷ 2  ≡ X (mod 4) does not have a solution, since there is no X such that 2 × X ≡ 1 (mod 4). When the modulus N is not prime, it is possible to divide a class represented by r by a class represented by s if and only if s and N are relatively prime (that is, if their only common factor is the number 1). For example, 7 ÷ 4 ≡ 4 (mod 9) since 4 × 4 ≡ 7 (mod 9)—in this case, 7 and 9 are relatively prime.

John L. Berggren
 

* * *


Universalium. 2010.

Look at other dictionaries:

  • 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. The Swiss mathematician Leonhard Euler pioneered the modern… …   Wikipedia

  • modular arithmetic — noun : arithmetic that deals with whole numbers where the numbers are replaced by their remainders after division by a fixed number in a modular arithmetic with modulus 5, 3 multiplied by 4 is 2 5 hours after 10 o clock is 3 o clock because… …   Useful english dictionary

  • modular arithmetic — noun Date: 1959 arithmetic that deals with whole numbers where the numbers are replaced by their remainders after division by a fixed number < in a modular arithmetic with modulus 5, 3 multiplied by 4 is 2 > …   New Collegiate Dictionary

  • Modular exponentiation — is a type of exponentiation performed over a modulus. It is particularly useful in computer science, especially in the field of cryptography. Doing a modular exponentiation means calculating the remainder when dividing by a positive integer m… …   Wikipedia

  • Modular multiplicative inverse — The modular multiplicative inverse of an integer a modulo m is an integer x such that That is, it is the multiplicative inverse in the ring of integers modulo m. This is equivalent to The multiplicative inverse of a modulo m exists if and only if …   Wikipedia

  • arithmetic — Synonyms and related words: Boolean algebra, Euclidean geometry, Fourier analysis, Lagrangian function, algebra, algebraic geometry, analysis, analytic geometry, associative algebra, binary arithmetic, calculation, calculus, ciphering, circle… …   Moby Thesaurus

  • Arithmetic function — In number theory, an arithmetic (or arithmetical) function is a real or complex valued function ƒ(n) defined on the set of natural numbers (i.e. positive integers) that expresses some arithmetical property of n. [1] An example of an arithmetic… …   Wikipedia

  • Modular form — In mathematics, a modular form is a (complex) analytic function on the upper half plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main… …   Wikipedia

  • Modular curve — In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half plane H by the action of a congruence subgroup Γ of the modular group of …   Wikipedia

  • Arithmetic mean — In mathematics and statistics, the arithmetic mean, often referred to as simply the mean or average when the context is clear, is a method to derive the central tendency of a sample space. The term arithmetic mean is preferred in mathematics and… …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”