Modular arithmetic | Binary operations
In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer a is an integer x such that the product ax is congruent to 1 with respect to the modulus m. In the standard notation of modular arithmetic this congruence is written as which is the shorthand way of writing the statement that m divides (evenly) the quantity ax β 1, or, put another way, the remainder after dividing ax by the integer m is 1. If a does have an inverse modulo m, then there are an infinite number of solutions of this congruence, which form a congruence class with respect to this modulus. Furthermore, any integer that is congruent to a (i.e., in a's congruence class) has any element of x's congruence class as a modular multiplicative inverse. Using the notation of to indicate the congruence class containing w, this can be expressed by saying that the modulo multiplicative inverse of the congruence class is the congruence class such that: where the symbol denotes the multiplication of equivalence classes modulo m.Written in this way, the analogy with the usual concept of a multiplicative inverse in the set of rational or real numbers is clearly represented, replacing the numbers by congruence classes and altering the binary operation appropriately. As with the analogous operation on the real numbers, a fundamental use of this operation is in solving, when possible, linear congruences of the form Finding modular multiplicative inverses also has practical applications in the field of cryptography, i.e. public-key cryptography and the RSA algorithm. A benefit for the computer implementation of these applications is that there exists a very fast algorithm (the extended Euclidean algorithm) that can be used for the calculation of modular multiplicative inverses. (Wikipedia).
Number Theory | Modular Inverses: Example
We give an example of calculating inverses modulo n using two separate strategies.
From playlist Modular Arithmetic and Linear Congruences
The modular inverse via Gauss not Euclid
We demonstrate a lesser-known algorithm for taking the inverse of a residue modulo p, where p is prime. This algorithm doesn't depend on the extended Euclidean algorithm, so it can be learned independently. This is part of a larger series on modular arithmetic: https://www.youtube.com/pl
From playlist Modular Arithmetic Visually
Inverse Matrices & Matrix Equations 4 Ex Multiplicative Inverses Full Length
I start by defining the Multiplicative Identity Matrix and a Multiplicative Inverse of a Square Matrix. I then work through three examples finding an Inverse Matrix. Inverse of 2 x 2 Matrix at 5:14 and 14:50 Inverse of a 3 x 3 Matrix at 21:32 Matrix Equation example at 39:58 Check out
From playlist Linear Algebra
Number Theory | Inverses modulo n
We give a characterization of numbers which are invertible modulo n.
From playlist Modular Arithmetic and Linear Congruences
Linear Algebra - Lecture 23 - The Inverse of a Matrix
In this lecture, we'll learn about the multiplicative inverse of a matrix. We'll discuss inverses of 2x2 matrices, as well as properties of inverses.
From playlist Linear Algebra Lectures
Review of Multiplicative Inverses
In this video we connect and review the ideas of multiplicative inverses and reciprocals
From playlist Middle School This Year
7B Inverse of a Matrix-YouTube sharing.mov
An introduction to the inverse of a square matrix.
From playlist Linear Algebra
From playlist ck12.org Algebra 1 Examples
Multiplicative Inverse and Reciprocals
http://www.youtube.com/view_play_list?p=8E39E839B4C6B1DE
From playlist Common Core Standards - 6th Grade
Modular Inverses, Generators, and Order: Linking Elementary Number Theory and Abstract Algebra
Solution to the unique number question posed at 5:01 We show two solution: one is more informal using intuition, and the other using congruences 1) Using Intuition We use a proof by contradiction: suppose that you can reach the same number twice WITHOUT first cycling back to 0. But since y
From playlist Summer of Math Exposition Youtube Videos
How does cryptography ACTUALLY work?
In this video I'll attempt to introduce you to some of the maths behind modern cryptography, which is in a sense how the world around us works now. Surprisingly, it has a lot to do with the simple ideas of division and remainders. We'll cover modular arithmetic basics, continued fractions
From playlist Summer of Math Exposition Youtube Videos
Cryptography_Modular_Arithmetic (Lecture 3)
From playlist Crypto1
An introduction to Modular Arithmetic, Lagrange Interpolation and Reed-Solomon Codes. Sign up for Brilliant! https://brilliant.org/vcubingx Fund future videos on Patreon! https://patreon.com/vcubingx The source code for the animations can be found here: https://github.com/vivek3141/videos
From playlist Other Math Videos
Math for Liberal Studies - Lecture 3.8.1 Affine and Multiplicative Ciphers
This is the first video lecture for Math for Liberal Studies, Section 3.8: More Modular Arithmetic and Public-Key Cryptography. In this lecture, I talk about how we can use multiplication in modular arithmetic to construct new ciphers. I also discuss the difficulty in finding the decryptio
From playlist Math for Liberal Studies Lectures
Discrete Structures: Modular arithmetic
A review of modular arithmetic. Congruent values; addition; multiplication; exponentiation; additive and multiplicative identity.
From playlist Discrete Structures
Why RSA encryption actually works (no prior knowledge required)
In this video, I am going to show you why RSA encryption works. I will prove the correctness of RSA from scratch, so no prior knowledge will be required. All results from number theory needed to understand why RSA works will be proven along the way. 00:00 1. Introduction, outline and disc
From playlist Cryptography
Modular Arithmetic in Mathematica & the Wolfram Language
ππΌππ ππππ? https://snu.socratica.com/mathematica To be notified of when our first Pro Course "Mathematica Essentials" is available, join our mailing list at: https://snu.socratica.com/mathematica In this video, we introduce modular arithmetic and how it can be visualized as "circular arit
From playlist Mathematica & the Wolfram Language
Modular Forms | Modular Forms; Section 1 2
We define modular forms, and borrow an idea from representation theory to construct some examples. My Twitter: https://twitter.com/KristapsBalodi3 Fourier Theory (0:00) Definition of Modular Forms (8:02) In Search of Modularity (11:38) The Eisenstein Series (18:25)
From playlist Modular Forms
About the Andr\'e-Oort Conjecture - Umberto Zannier
Umberto Zannier Scuola Normale Superiore de Pisa, Italy May 12, 2010 For more videos, visit http://video.ias.edu
From playlist Mathematics