Why Is Modular Arithmetic Important in Math & Everyday Life?
Modular arithmetic is a topic that will come under number theory, which roughly speaking is the study of integers and their properties. Modular arithmetic basically calculates the power of remainders when we solve problems. In modular arithmetic, the numbers that we deal with are integers and the operations that we use are addition, subtraction, multiplication and division. The main difference between modular arithmetic and the basic arithmetic we learned is that in modular arithmetic all operations are performed regarding a positive integer, i.e. the modulus.
It is a special type of arithmetic that consists of only integers. The objective of this article is to explain the basics of modular arithmetic and modular congruence. We will also understand the modular arithmetic formula and its various applications.
Modular Arithmetic Definition
In its most elementary form, Modular arithmetic is sometimes referred to as modulus arithmetic or clock arithmetic. We do arithmetic just like counting that resets to zero every time after a certain whole number N greater than one, known as the modulus (mod), has been reached.
Application of modular arithmetic is widely used in the field of computer science and cryptography.
Let n be a positive integer. We denote the set [0,1…(n-1)] by Zn .
Consider two integers x,y to be the same if x and y differ by a multiple of n, and we write this as x=y(mod n) and say that x and y are congruent modulo n. We may omit (mod n) when it is clear from the context. Every integer x is congruent to some y in Zn . When we add or subtract multiples of n from an integer x to reach some yZn , we can say that we are reducing x modulo n, and the value of y is the residue.
Modular Arithmetic Rules
Modular arithmetic follows the same rules as classical arithmetics follow.
Given below modular arithmetic rules
Suppose a,b,c and d are integers and m is a positive integer
Addition Rule
If \[a\equiv b (mod\ m)\] and
\[c\equiv d (mod\ m)\] , then
\[a + c \equiv b + d (mod\ m)\]
Modular Subtraction Rule
If \[a\equiv b (mod\ m)\] and
\[c\equiv d (mod\ m)\], then
\[a - c \equiv b - d (mod\ m)\]
Modular Arithmetic Multiplication Rule
If \[a\equiv b (mod\ m)\] and
\[c\equiv d (mod\ m)\], then
\[a \times c \equiv b \times d (mod\ m)\]
Given below are addition and multiplication modulo n satisfy the following properties:
(x + y) + z = x + (y + z)
(x. y)z = x(y. z)
x + 0 = 0 + x = x
1. x = x. 1
x + (n - x) = (n- x) + x = x
x + y = y + x
x. y = y. x
x. 0 = 0. x
(x + y) + z = x + (y + z)
(x. y)z = x(y. z)
x + 0 = 0 + x = x
1. x = x. 1
x + (n - x) = (n- x) + x = x
x + y = y + x
x. y = y. x
x. 0 = 0. x
Modular Multiplication
Here we have provided modular multiplication rules for two and three elements
\[(a \times b) mod\ m = ((a\ mod\ m) \times (b\ mod\ m) mod\ m)\]
\[(a \times b \times c) mod\ m = ((a\ mod\ m) \times (b\ mod\ m) (c\ mod\ m) mod\ m)\]
The same property we can apply for more than three numbers.
We will understand this with an example:
Example: Find the remainder of 151719 when we divide it by 7.
Solution: Here the numbers are 15,17 and 19. First, we will calculate the individual remainder.
If we divide 15 by 7 we will get 1 as the remainder.
If we divide 17 by 7 we will get 3 as the remainder.
If we divide 19 by 7 we will get 5 as the remainder.
Hence, the remainder of the expression \[\frac{(15\times 17\times 19)}{7}\] will be equal to \[\frac{(1\times 3\times 5)}{7}\].
So, the combined remainder will be equal to the remainder of \[\frac{15}{7}\] i.e. 1.
Division in Modular Arithmetic
We can define modular division only when the modular inverse of the divisor exists. The inverse of an integer ‘x’ is another integer ‘y’ such that it should follow (x*y) % m = 1 where m is known as the modulus.
Consider two number two number 5 and 3
If we have to find the value of 5 mod 3 the value will be 2. It means 2 is the remainder when we divide 5 by 3.
Application of Modular Arithmetic
Modular arithmetic is an extremely flexible problem-solving tool. The following topics are a few applications and extensions of its use:
Divisibility rules
Linear congruence
In Modular arithmetic in cryptography
Public encryption key
Private decryption key
Conclusion:
Consider four integers a,b,c,d and a positive integer m such that \[a\equiv b (mod\ m)\] and \[c\equiv d (mod\ m)\] In modular arithmetic, it holds the following identities
Addition \[a + c \equiv b + d (mod\ m)\]
Subtraction \[a - c \equiv b - d (mod\ m)\]
Multiplication \[a \times c \equiv b \times d (mod\ m)\]
Division \[\frac{a}{e} = \frac{b}{e}(mod \frac{m}{gcd(m,e)})\] where e is a [positive integer that divides a and b
Exponential form \[a^{e} \equiv b^{e} (mod\ m)\] where e is a positive integer.
FAQs on Modular Arithmetic Made Easy: Rules, Tricks & Uses
1. What exactly is modular arithmetic?
Modular arithmetic is a system of arithmetic for integers, often called 'clock arithmetic,' where numbers 'wrap around' upon reaching a certain value—the modulus. For example, on a 12-hour clock, the hours go from 1 to 12 and then start over. In modular arithmetic, we are only interested in the remainder after division by the modulus.
2. How is congruence in modular arithmetic expressed?
Congruence is expressed with the formula a ≡ b (mod n). This means that two integers, 'a' and 'b', are congruent modulo 'n' if they have the same remainder when divided by 'n'. An equivalent way to understand this is that their difference (a - b) is an integer multiple of 'n'.
3. What does the expression 17 mod 5 mean?
The expression 17 mod 5 asks for the remainder when 17 is divided by 5. To calculate this, you divide 17 by 5, which gives a quotient of 3 and a remainder of 2. Therefore, 17 mod 5 = 2. In congruence notation, this is written as 17 ≡ 2 (mod 5).
4. Why is an expression like 7 mod 10 equal to 7?
This is a fundamental concept for beginners. When the first number (the dividend) is smaller than the modulus, the remainder is simply the first number itself. When you divide 7 by 10, the quotient is 0 and the remainder is 7. So, 7 mod 10 = 7. The number 'doesn't wrap around' the modulus yet.
5. What are the key properties of modular arithmetic operations?
Modular arithmetic has consistent properties for addition, subtraction, and multiplication:
- Addition: (a + b) mod n = [(a mod n) + (b mod n)] mod n
- Subtraction: (a - b) mod n = [(a mod n) - (b mod n)] mod n
- Multiplication: (a * b) mod n = [(a mod n) * (b mod n)] mod n
These properties allow us to work with smaller numbers (the remainders) first, which simplifies calculations involving very large numbers.
6. How does modular arithmetic differ from regular arithmetic?
The main difference is the set of numbers used. Regular arithmetic operates on an infinite set of integers. Modular arithmetic, however, operates on a finite set of integers from 0 to (n-1), where 'n' is the modulus. It's a 'wrapping around' system, like a clock, rather than a continuous number line. This finite nature is what makes it uniquely powerful for specific applications.
7. What are some real-world examples of modular arithmetic?
Modular arithmetic is used in many everyday and technological contexts:
- Time Calculations: A 12-hour clock is a modulo 12 system.
- Calendars: The days of the week operate on a modulo 7 system.
- Cryptography: It forms the basis for encryption methods like the RSA algorithm, securing online communications.
- Computer Science: Used in hash functions, pseudorandom number generation, and checking for errors (e.g., ISBN numbers).
8. Why is modular arithmetic so important in the field of cryptography?
Its importance in cryptography comes from the creation of 'one-way functions.' These are calculations that are easy to perform in one direction but extremely difficult to reverse. For example, multiplying two large prime numbers (a modular multiplication) is fast. However, taking the resulting product and finding the original prime factors (the reverse operation) is computationally infeasible. This principle underpins public-key cryptography, where the 'public key' is based on the easy operation and the 'private key' is protected by the difficulty of the reverse operation.
9. How do you calculate the modulus for a negative number, like -15 mod 4?
To find the modulus for a negative number, you need to find the smallest non-negative remainder. You can do this by adding multiples of the modulus to the negative number until you get a positive result. For -15 mod 4, you add multiples of 4:
- -15 + 4 = -11
- -15 + 8 = -7
- -15 + 12 = -3
- -15 + 16 = 1
The first non-negative result is 1. Therefore, -15 ≡ 1 (mod 4).
10. Does division exist in modular arithmetic?
Standard division is not directly defined in the same way as regular arithmetic. Instead, we use a concept called the modular multiplicative inverse. Dividing by a number 'a' is equivalent to multiplying by its modular inverse. The inverse of 'a' (mod n) is an integer 'b' such that (a * b) ≡ 1 (mod n). This inverse only exists if 'a' and 'n' are coprime (their greatest common divisor is 1).
