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Authors:
William Stallings
Chapter:
Introduction To Number Theory
Exercise:
Review Questions
Question:3 | ISBN:9781292158587 | Edition: 7

Question

What is the difference between modular arithmetic and ordinary arithmetic?

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Answer

Modular arithmetic, also known as clock arithmetic or arithmetic modulo, is a specialized form of arithmetic that deals with numbers within a fixed range or modulus. It operates on a set of integers and involves performing addition, subtraction, multiplication, and division operations, but with a unique twist.

  • The main difference between modular arithmetic and ordinary arithmetic lies in how the operations are performed and how the numbers are treated.
  • In ordinary arithmetic, numbers extend indefinitely in both the positive and negative directions. However, modular arithmetic confines numbers to a specific range or modulus.

 

Here are some key differences:

  • Modulus: In modular arithmetic, a modulus is defined, which determines the range within which numbers operate. It is typically denoted by "m" and represents the number of values in the set. For example, if the modulus is 12, the numbers in the set range from 0 to 11. In ordinary arithmetic, there is no specific modulus, and numbers can be infinitely large or small.

  • Wrapping around: In modular arithmetic, when a result exceeds the modulus, it "wraps around" back to the beginning of the set. For example, in modulo 12 arithmetic, 10 + 5 = 3 because 10 + 5 = 15, but 15 modulo 12 equals 3. In ordinary arithmetic, there is no wrapping around, and the result can be any number.

  • Unique inverses: In modular arithmetic, every nonzero number has a unique multiplicative inverse within the given modulus. This means that for every number "a," there exists another number "b" such that a * b is congruent to 1 modulo m. In ordinary arithmetic, not every number has a unique multiplicative inverse.

  • Operations: In modular arithmetic, addition, subtraction, multiplication, and division are performed as usual, but with the additional step of taking the modulus of the result. This ensures that the result stays within the range defined by the modulus. In ordinary arithmetic, there is no need to take a modulus unless specifically required.

 

Modular arithmetic finds applications in various fields, including computer science, cryptography, number theory, and digital signal processing. It provides a useful framework for dealing with periodic or cyclical phenomena and allows for simplified calculations in certain contexts.

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