Modular Arithmetic in Mathematica & the Wolfram Language
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- Опубліковано 8 чер 2024
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In this video, we introduce modular arithmetic and how it can be visualized as "circular arithmetic" or "clock arithmetic." Mathematica has many built-in functions that help you do modular arithmetic, including the Mod function, the PowerMod function, the ModularInverse function, and many more. We show several examples to help you get started.
Thank you to our VIP Patreon Members who helped make this video possible! Robert Fulbright, Marko Popovic, Scott Dreblow, Marcos Silveira, Christopher Kemsley, Philip Rice, Eric Eccleston, Jeremy Shimanek, Michael Shebanow, Kavon Johnson, Alvin Khaled, Kevin B, John Krawiec, Umar Khan, MdeG, Carlos Araujo, and Tracy Karin Prell.
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MATHEMATICA ESSENTIALS by Socratica
Modular Arithmetic
0:00 Intro to Modular Arithmetic
2:17 Mod function
3:21 Number Theory/ Fermat's Little Theorem
3:43 Euler totient function
4:31 PowerMod function
5:41 ModularInverse function
6:34 Using Solve with modular equations
7:34 PolynomialMod function
7:46 PrimitiveRoot & PrimitiveRootList functions
7:58 EulerPhi
8:09 MultiplicativeOrder function
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Written & Produced by Michael Harrison & Kimberly Hatch Harrison
Edited by Megi Shuke
About our Instructors:
Michael earned his BS in Math from Caltech, and did his graduate work in Math at UC Berkeley and University of Washington, specializing in Number Theory. A self-taught programmer, Michael taught both Math and Computer Programming at the college level. He applied this knowledge as a financial analyst (quant) and as a programmer at Google.
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Mathematica is a *Beautiful* way to find solutions! Thank you for creating this wonderful video!
Thank you for your kind words, Tracy!
The 3rd argument to Mod[] is worth mentioning: the lower bound of the result range. E.g., if you want the representative range around 0, Mod[a, b, -b/2] does the trick:
Table[Mod[n, 7, -3], {n, 0, 7}] ⇒ {0, 1, 2, 3, -3, -2, -1, 0}
7:22: Ughhh... :) Ctrl+L copies closest input cell's contents from above and inserts at point if inside a cell, or to a new input cell if the point is between cells.
This is a terrific series!
Very helpful video
Great 👍
1:24 is this explanation correct for 1(mod 7). 34 divided by 7 gives 6 remainders. I am confused.
Correct, 34 ≡ 6 (mod 7), both 6 and 34 are in the same congruency class, the last line.
0:13 Why do some people put arrows at both ends? 🤔 The arrow is there to point in the direction of increase. (And, no, it is not an exclusively american thing; I went to uni in the US and my profs did not do this, but I have seen brits and south-asians on YT who do).
Interesting! In general the arrowheads indicate the line continues in both directions. With this interpretation, the interval (0, infinity) would only point to the right while (-infinity, 0) would only point to the left.
@@Socratica It’d be interesting to find the source of this usage. In the usage I’m used to (with the arrow indicating the direction of increase, obviously in use in the western world in the late 1900s since I am Swedish educated in Sweden and the US 😊), a number line is infinite by default; to show that you only want to consider part of it you have to use special symbols for intervals, like filled or open circles.
I guess I have to re-watch this again and again to better understand it.
It's amazing how sometimes it takes a few days and suddenly it becomes clear! Maybe our brains are puzzling it out in the background in the meantime.
First :)
First reply!
Second @Battery
Like a negative post on a battery @Battery.
All good.
Positive!
Excellent ☝️take your like👍