Modular arithmetic victor adamchik fall of 2005 plan 1. Modular arithmetic is a fundamental component of cryptography. In modular arithmetic, the numbers we are dealing with are just integers and the operations used are addition, subtraction, multiplication and division. In modular arithmetic, which has some interesting features for understanding the properties of numbers, one is interested only in the remains, and this part of mathematics deals exclusively with integer numbers. What is modular arithmetic introduction to modular. Modular arithmetic and integer representations unsigned, signmagnitude, and twos complement representation. The only important property of 3 in the proof of theorem 4. The calculator below solves a math equation modulo p. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers wrap around when reaching a certain value, called the modulus. If nis an integer, two integers are equal mod nif they di. Modular arithmetic is often tied to prime numbers, for instance, in wilsons theorem, lucass theorem, and hensels lemma, and generally appears in fields like cryptography, computer science, and computer algebra. The ordinary base b logarithm of c is defined to be the real number x such that.
The swiss mathematician leonhard euler pioneered the modern approach to. Instead of hitting you in the face with formulas, lets explore an idea weve been subtly. In this sense, 1 5 9 1 mod 4, since they all leave the same remain, 1, when divided by 4. Everything you need to know about modular arithmetic. We say that two integers a and b are congruent modulo m if b. Modular arithmetic is also called clock arithmetic because we only use 12 numbers to tell standard time. In this lecture, i will quickly go over the basics of the subject and. It is important to check that it does not matter which numbers we are choosing from.
Mar 04, 2011 in other words, modular arithmetic is a method for doing addition, subtraction, multiplication, and division with integers where the numbers curve around the number line cyclically instead of continuing on forever. When we add or subtract multiples of n from an integer x to reach some y. This clearly proves that no common divisor to a and b is greater than gcda. In contrast to subjects such as arithmetic and geometry, which proved useful in everyday problems in commerce and architecture, as.
There are several useful properties of modular arithmetic. Informally we may think of two numbers as congruent modulo n when they have the same remainder on division by n. The intersection of the row headed 7 and the column headed 11 gives the number 6. It turns out that modular arithmetic follows many of the same rules of classical arithmetic, thus making it very easy to work with. Anybody can master modular arithmetic with a little bit of practice. Though it seems that arithmetic on zn is the same as on z, do not be deceived. The modular logarithm seems to resemble the ordinary, nonmodular logarithm. Modular arithmetic sounds like a mathematical subject that will make your head spin. Modular arithmetic has been applied to fields ranging from number theory to computer. The euclidean algorithm is a very well know algorithm to find the. Modular 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. Mathematics stack exchange is a question and answer site for people studying math at any level and professionals in related fields. On clocks we start at 1 instead of 0, continue to 12, and then start at 1 again.
Modular arithmetic is something we use everyday when we look at a clock. In other words, unless we add two numbers and get a number that is 5 or larger, everything works like before. This goal of this article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic. Basically, it is a kind of integer arithmetic that reduces all numbers to ones that belongs to a fixed set 0. I hadnt given it much thought, but realized the modulo is extremely powerful. In this video, i explain the basics of modular arithmetic with a few simple examples. Congruence modulo if youre seeing this message, it means were having trouble loading external resources on our website. Modular arithmetic and elementary algebra 1 euclids algorithm. Properties of multiplication in modular arithmetic. High school mathematics extensionsprimesmodular arithmetic. How modular logarithms differ from ordinary logarithms. Jul 24, 2014 modular arithmetic is a fundamental component of cryptography.
An intuitive usage of modular arithmetic is with a 12hour clock. You will get even more practice on a quiz when youve finished the lesson. Dec 24, 2014 modular arithmetic properties december 24, 2014 june 7, 2015 vamsi sangam in competitive programming, modular arithmetic is an essential tool in solving big number problems. We say \a is congruent to b modulo m and write a b mod m if m ja b i. Modular arithmetic is a special type of arithmetic that involves only integers. You may have worked a lot to get the logic, but the output must be given as they say. So instead of introducing finite fields directly, we first have a look at another algebraic structure. Modular arithmetic and elementary algebra 1 euclids. Modular arithmetic is important in number theory, where it is a fundamental tool in the solution of diophantine equations particularly those restricted to integer solutions. You may also enter other integers and the following modular operations. Modular arithmetic lets us carry out algebraic calculations on integers with a systematic disregard for terms divisible by a certain number called the modulus. In some ways this is a generalization of the concept of parity. The modern approach to modular arithmetic was developed by carl friedrich gauss in his book disquisitiones arithmeticae, published in 1801.
Now that you know fermat s little theorem, solve problem 3 in a. Modular arithmetic has been a major concern of mathematicians for at least 250 years, and is still a very active topic of current research. In competitive programming, modular arithmetic properties are essential tools in solving big number problems. The best preparation for the twotier modular specification and additional support for the 2007 specification 2381with ten years experience of publishing for edexcel mathematics and as a preferred partner of edexcel, we can offer the best option for the 2006 twotier modular specification.
The length of the circular number line in modular arithmetic is called the modulus. Modular arithmetic properties congruence, addition, multiplication, proofs. Definition let m 0 be a positive integer called the modulus. Inverses in modular arithmetic we have the following rules for modular arithmetic. An equivalence class consists of those integers which have the same remainder on division by n. This kind of \reduced algebra is essential background for the mathematics of. The modular arithmetic properties allow me to solve this equation the way i would solve a linear equation, up to a point. Because you already know how to do modular arithmetic even if youve never heard of it before. In fact, many students apply modular arithmetic every. The modulo arithmetic lesson aims to teach students addition and. In the problem statement, whenever they say, print the answer, its not that simple.
The only difference between modular arithmetic and the arithmetic you learned in your primary school is that in modular arithmetic all operations are performed regarding a positive integer, i. This equation reminds one of the modular equation defining the modular logarithm. In essence, everything we are familiar with for usual addition works for modular arithmetic aslongasitmakessense. Examples are a digital clock in the 24hour system, which resets itself to 0 at midnight n. Indeed, euclids algorithm also allows to nd such integers s and t. In this article, i will explain what modular arithmetic is, illustrate why it is of importance for mathematicians, and discuss some recent breakthroughs. The mod n is sometimes omitted when it is clear from the context.
Read and learn for free about the following article. Apr 22, 2018 congruence, modular arithmetic, 3 ways to interpret a. First, there is the fact that congruence modulo n satis es 3 popular properties of relations. But when n is a prime number, then modular arithmetic keeps many of the nice properties we are used to with whole numbers.
Modular arithmetic, cryptography, and randomness for hundreds of years, number theory was among the least practical of mathematical disciplines. Let us learn more about this modulo operator and its properties. Math circle thursday january 22, 2015 what is modular arithmetic. Number theory, in turn, is foundational for cryptography, the science of keeping ones communications and data secure from eavesdropping third parties. Modular arithmetic is a form of arithmetic dealing with the remainders after integers are divided by a fixed modulus m. In fact, the term draws images of a math nerd scribbling foreign symbols on a blackboard, but in reality it isnt that difficult. Modulo is a mathematical jargon that was introduced into mathematics in the book disquisitiones arithmeticae by carl friedrich gauss in 1801. Modulo a prime number we have seen that modular arithmetic can both be easier than normal arithmetic in how powers behave, and more di. Note that you can use the second property and induction to show that if a b mod m, then an bn mod m for all n. In order to highlight what is going on, we try to compare and contrast modular. The equivalence classes are also known as congruence classes modulo n. Congruence, modular arithmetic, 3 ways to interpret a. For any positive integers, there exist integers s and t such that gcda.
If youre seeing this message, it means were having trouble loading external resources on our website. Modular arithmetic an overview sciencedirect topics. The calculator performs arithmetic operations modulo p. For almost all its history, the study of modular arithmetic has been driven purely by its inherent. The length of the circular number line in modular arithmetic is. The best way to introduce modular arithmetic is to think of the face of a clock. I will strongly encourage you to read all the topics in this section. If youre behind a web filter, please make sure that the domains. Generalizations of the subject led to important 19thcentury attempts to prove fermats last theorem and the development of significant parts of modern algebra. If we do get a number that is 5 or larger, we just subtract 5, and proceed. Introduction integer congruences department of mathematics. Groups, modular arithmetic and finite fields the structure of a finite field is a bit complex. In fact, this is the example we shall need for the rsa cryptosystem. Followup exercise read up on euclids algorithm for.
A reader recently suggested i write about modular arithmetic aka taking the remainder. We know that a a 0, and one of the elementary results seen previously is that n j0 for any integer n. The modulo arithmetic lesson aims to teach students addition and multiplication modulo n. Dec 24, 2014 in competitive programming, modular arithmetic properties are essential tools in solving big number problems.
Zn, we say are reducing x modulo n, and y is the residue. In some sense, modular arithmetic is easier than integer artihmetic because there are only finitely many elements, so to find a solution to a problem you can always try every possbility. Two integers a and b are congruent modulo m if they di. Modular arithmetic properties theory of programming. In modular arithmetic, we select an integer, n, to be our \modulus.
Thus, modular arithmetic gives you another way of dealing with divisibility relations. I multiply out the left side, then get the xs on one side. In this lesson, well look at some rules and properties of addition, subtraction, and multiplication in modular arithmetic. If none of them work then we know \y\ does not have an inverse. Rather than say the integers a and b are equivalent we say that they are congruent modulo n. Divisibility and modular arithmetic are foundational to number theory, which is the theory of the integers. Applications of modular arithmetic hashing, pseudorandom numbers, ciphers.
Notice that the definition is the same as for ordinary subtraction of whole numbers. Applications of modular arithmetic are given to divisibility tests and to block ciphers in cryptography. Enter an integer number to calculate its remainder of euclidean division by a given modulus. Modular arithmetic lets us carry out algebraic calculations on integers with a. Modular arithmetic given a positive integer n, and two integers a and b, we say \a is congruent to b modulo n and write \a b mod n i a b is a multiple of n or equivalently i n divides a b. Modular arithmetic highlights the power of remainders when solving problems.