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In mathematics, modular arithmetic is a system of
arithmetic for certain equivalence classes of integers, called congruence classes. Sometimes it is suggestively called
'clock arithmetic', where numbers 'wrap around' after they reach a
certain value (the modulus). For example, when the modulus is 12, then any two numbers that leave the same
remainder when divided by 12 are equivalent (or "congruent") to each other. The numbers
- ..., −34, −22, −10, 2, 14, 26, ...
are all "congruent modulo 12" to each other, because each leaves the same remainder (2) on division by 12. The collection of all such numbers is a congruence class.
As explained below, one can add such congruence classes to get another such congruence class, subtract two such classes to get
another, and multiply such classes to get another. When the modulus is a prime
number, one can always divide by any class not containing 0.
Definition of modulo
Two discrepant conventions prevail:
- the one originally introduced by Gauss two centuries
ago, still used by mathematicians, and suitable for theoretical mathematics, and
- a newer one adhered to by computer scientists and perhaps more suitable for computing.
The older convention, used by mathematicians
The original convention is that the expression
-
means that a and b are both in the same "congruence class" modulo n, i.e., both leave the same
remainder on division by n, or, equivalently, a − b is a multiple of n. Thus we have,
for example
-
since 63 and 83 both leave the same remainder (3) on division by 10, or, equivalently, 63 − 83 is a multiple of 10. One
says:
- "63 is congruent to 83, modulo 10,"
or
- "63 and 83 are congruent to each other, modulo 10."
"Modulo" is usually abbreviated to "mod" in speaking, just as in writing. The parentheses, i.e., the round brackets (), are
usually not written, but in this case they emphasize the difference between the traditional mathematical convention and the
modern computing convention. The mathematical usage parses the phrase differently from the computing usage.
In Latin, the language in which Gauss wrote, modulo is the ablative case of
modulus. The number n, which in this example is 10, is the modulus.
The newer convention, used in computing
According to the newer convention, in general, a mod n is the remainder on integer division of a by
n.
For instance, 63 mod 10 = 3, the remainder left when 63 is divided by 10. As another example, 5.5 mod 1 = 0.5, the remainder
when 5.5 is divided by 1. Similarly, 23 mod 12 = 11. (Calculations mod 12 are what one does when converting the time from a 24
hour clock to a 12 hour clock.)
Remainder after ordinary division is insufficient when one of the operands is negative. Consider these cases:
- −63 mod -10 = −3
- −63 mod 10 = 7
- 63 mod −10 = −7
The difference in conventions is not very serious, in fact; it is reasonably thought of as reflecting the preference, for
computational purposes, of a normal form over the underlying equivalence relation. This can be regarded mainly as a notational
convention in this case, where there is a strict-sense normal form.
Implementation of the 'mod' function
Some calculators have a mod() function button.
Some programming languages have a mod() function or
similar, such as mod(x,y) in Perl. Some also support expressions that use
"%" as a 'mod' operator, as in x % y.
x mod y can be calculated by using equations, in terms of other functions. Differences may arise according
to the scope of the variables, which in common implementations is broader than in the definition just given.
In terms of the floor function floor(z), the greatest integer less than or equal to z:
- mod(x,y) = x − y × floor(x ÷ y)
This definition allows for x and y to be typed as integers or rational numbers.
The expression x mod 0 is undefined in the majority of numerical systems, although some do define it to be
x.
Applications of modular arithmetic
Modular arithmetic, first systematically studied by Carl
Friedrich Gauss at the end of the eighteenth century, is applied in number theory, abstract algebra, cryptography, and visual and musical art.
The fundamental arithmetic operations performed by most computers are actually modular arithmetic, where the modulus is
2b (b being the number of bits of the values being operated on). This comes to light in the
compilation programming languages such as C; where for
example arithmetic operations on "int" integers are all taken modulo 232, on most computers.
In art
In music, because of octave and enharmonic equivalency (that is, pitches in a 1/2 or 2/1 ratio are equivalent, and C# is the same as Db),
modular arithmetic is used in the consideration of the twelve tone equally tempered scale, especially in twelve tone music. In visual art modular arithmetic can be used to create artistic patterns based on
the multiplication and addition tables modulo n (see link below).
Some consequences of the mathematical usage
Recall from above that two integers a, b congruent modulo n, written as
- a ≡ b (mod n) if their difference a − b is divisible by n, i.e. if a − b = kn for some
integer k.
Using this definition, we can generalize to non-integral moduli. For instance, we can define a ≡ b
(mod π) if a − b = kπ for some integer k. This idea is
developed in full in the context of ring theory below.
Here is an example of the congruence notation.
- 14 ≡ 26 (mod 12).
This is an equivalence relation, and the equivalence class of the integer a is denoted by
[a]n (or simply [a] if the modulus n is understood.) Other notations include
a + nZ or a mod n. The set of all equivalence classes is denoted
Z/nZ = { [0]n, [1]n, [2]n,
..., [n-1]n }.
If a and b are integers, the congruence
- ax ≡ b (mod n)
has a solution x if and only if the greatest
common divisor (a, n) divides b. The details are recorded in the linear congruence theorem. More complicated simultaneous
systems of congruences with different moduli can be solved using the Chinese remainder theorem or the method of successive substitution.
This equivalence relation has important properties which follow immediately from the definition: if
- a1 ≡ b1 (mod n) and
a2 ≡ b2 (mod n)
then
- a1 + a2 ≡ b1 + b2 (mod
n)
and
- a1a2 ≡ b1b2 (mod
n).
This shows that addition and multiplication are well-defined operations
on the set of equivalence classes. In other words, addition and multiplication are defined on
Z/nZ by the following formulae:
- [a]n + [b]n = [a + b]n
- [a]n[b]n = [ab]n
In this way, Z/nZ becomes a commutative ring with n elements. For instance, in the ring Z/12Z, we
have
- [8]12[3]12 + [6]12 = [30]12 = [6]12.
In abstract algebra, it is realized that modular arithmetic is a
special case of forming the factor ring of a ring modulo an ideal. If R is a commutative ring, and I is an ideal of R, then the
elements a and b of R are congruent modulo I if a −
b is an element of I. As with the ring of integers, this turns out to be an equivalence relation, and addition
and multiplication become well-defined operations on the factor ring R/I.
In the ring of integers, if we consider the equation ax ≡ 1 (mod n), then we see that
a has a multiplication inverse if and only if a and n are coprime. Therefore, Z/nZ is a field if and only if n is prime.
It can be shown that every finite field is an extension of
Z/pZ for some prime p.
An important fact about prime number moduli is Fermat's little theorem: if p is a prime number and a is any integer, then
- ap ≡ a (mod p).
This was generalized by Euler: for any positive integer n
and any integer a that is relatively prime to n,
- aφ(n) ≡ 1 (mod n),
where φ(n) denotes Euler's φ function
counting the integers between 1 and n that are coprime to n.
Euler's theorem is a consequence of the Theorem of Lagrange,
applied to the group of units of the ring Z/nZ.
Another "computing" usage
An implied meaning of modulo in computing contexts is "valid up to this value." For example, "addition is modulo
1,000" means that the addition operation being described provides valid answers until the sum goes beyond 1,000.
Digital representations of number spaces are not infinite (see binary numeral systems). Thus, if a computer is representing a set of positive integers as 8-bits, the values that can be represented range from 0 to 255. When an addition (or multiplication,
or whatever) results in a number above this cutoff, the typical behavior is for the values to wrap around. For example, in the
8-bit positive integer situation, 255 + 1 = 0. This computer is therefore described as "modulo 256". Furthermore, some computers
do different operations with different bit representations. So although the storage of integers may be 8-bit ("modulo 256"), the
addition of integers may be 12-bit ("modulo 4096"), or anything else. Thus individual operations can also be described as "modulo
x".
In the case of signed (positive and negative) integers, or floating point numbers, the wrap around effect is more complicated,
and is not always perfectly analogous to the formal mathematical modulo. However, the slang persists such that "addition is
modulo 1000" may not literally mean (in fact cannot literally mean) that the computer does addition in log21000 bits, but may simply mean "watch out: if you go over 1000 this computer will give you weird
results".
More general use of the word modulo by mathematicians
To say that any two things are the same "modulo" a third thing means, more-or-less, that the difference between the first two
is accounted for or explained by the third. That is, the up to concept is often
talked about this way, using modulo as a term alerting the hearer. In mathematics, this admits various precise
definitions. In particular, two members of a ring or an algebra are congruent modulo an ideal if the difference between them is
in the ideal. The use of the term in modular arithmetic is a special case of that usage, and that is how this more general usage
evolved. Some loose terms such as almost all can in this way acquire
precise meanings from a Boolean algebra version, in which symmetric difference of sets replaced arithmetical subtraction; for
example "modulo finite sets".
Slang use of the word modulo
Mathematicians speaking of things non-mathematical still say "A is the same as B modulo C" when they mean A is the same as B
except for differences accounted for by C. But in such non-mathematical contexts, the phrase may not admit any very precise
definition. Consequently mathematicians and computer scientists often use the phrase in an informal or even jocular way.
Some users of the term either lack this theoretical viewpoint or else ignore it, and use the word "modulo" more-or-less
synonymously with the preposition except.
Examples
- "http and https are the same, modulo
encryption." - means "the only difference between http and https is the addition of encryption".
- "These two characters are equal."
- "You mean, equal modulo case." - indicates that the first speaker is wrong: the characters are not the same, one is uppercase
and the other lowercase.
- "The two students performed equally well on the exam, modulo some minor computational mistakes." - means that the two
students demonstrated an equal understanding of the material and its application, but one of them lost some points for minor
computational mistakes.
- "This code is finished modulo testing" - means "this code is finished except for testing". Since testing is generally
considered quite important, whereas in mathematics the use of modular arithmetic generally ignores the difference between
modulo-equal numbers, use of a phrase like this might be deliberate understatement.
External resources
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