Modular Math - Walton High

Modular Math
Jack Liu and Mark Duane
Mrs. Miller
Multivariable Calculus Spring
Intro to Modular Math
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Sometimes called clock math
Euler pioneered the modern approach
Consists of looking at a number and its
remainder when divided by other numbers
Notations
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Only one is commonly used ( A ≡ B
mod(n) )
Can also be denoted with a base,
I.e. base 2, base 8, base 7
If two numbers have the same
remainder it becomes (A = B
mod(n))
Modular, an Explanation
X ≡ A (mod B)
X/B, the remainder is A.
So: X-A=multiple of B
Simple isn't it :)
Examples
1. Is 4 congruent to 7 mod 6?
2. Is 4 congruent to 7 mod 3?
3. Write that 2 is true.
The Math
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To prove modularity
A-B or B-A has to be a multiple of n
Properties (for all numbers)
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Properties if A, B, and n are integers
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Examples
Find the remainder when 2012 is added to 3020, mod 4
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both numbers are evenly divisible by 4, so the remainder of
the addition is 0.
Find the remainder of 3116 + 1059 mod 7
Find the remainder when 6515*65105 mod 5
• since both are evenly divisible by 5 the remainder is 0
Find the remainder of 897*8416 mod 6
Your Turn
Compute if 6 is congruent to 9 mod 2.
Compute if 4 is congruent to 30 mod 2
In what base is 10 congruent to 13?
Residue Systems
A residue system is a set that contains one of each number that
has a different remainder of n
For Example, mod 4, some residue systems are:
* {1,2,3,4}
* {13,14,15,16}
* {27,32,37,42}
What is NOT a residue system?
*{1,5,6,7} 1 and 5 are congruent mod 4
Examples of residue Systems
Is _______ a residue system?
• {24, 31, 99, 58, 67, 21} base 10
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{201, 3002, 60008, 90000, 7004} mod 5
Write a residue system for mod 13 and no a simple
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} will not count, nor
anything simple based off that...
Uses, Chinese Remainder Theorem
The Chinese Remainder theorem is used to find the minimum
number that fulfills a set of requirements.
The common example is, a woman who tells a policeman that
she lost her basket of eggs, and that if she took three at a time
out of it, she was left with 2, if she took five at a time out of it
she was left with 3, and if she took seven at a time out of it she
was left with 2. She then asks the policeman what is the
minimum number of eggs she must have had.
The answer is 23
Chinese Remainder Continued
For example, consider the problem of finding an integer x such that
A brute-force approach converts these congruences into sets and writes the
elements out to the product of 3×4×5 = 60 (the solutions modulo 60 for each
congruence):
x ∈ {2, 5, 8, 11, 14, 17, 20, 23, 26, 29, 32, 35, 38, 41, 44, …}
x ∈ {3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, …}
x ∈ {1, 6, 11, 16, 21, 26, 31, 36, 41, 46, 51, 56, …}
To find an x that satisfies all three congruences, intersect the three sets to get:
x ∈ {11, …}
Which can be expressed as
Uses, Computers
All computers understand two states, on and off
The states are denoted as 1 and 0 respectively
With these two states we get Binary, counting in ones and
zeroes
Ex, 3 in binary is 11, 2 is 10 and so on
Newer Computers can process greater numbers
most can also handle Octal, base 8, and some can handle
Hexadecimal, Base 16
Uses, Computer, Hexidecimal
0–9 represent values zero to nine, as they normally do
A, B, C, D, E, F to represent values ten to fifteen.
A=10
B=11
C=12
D=13
E=14
F=15
For example, the hexadecimal number 2AF3 is equal, in
decimal, to (2 × 163) + (10 × 162) + (15 × 161) + (3 × 160), or
10,995.
Representing numbers in different
bases
This is used in computing when reducing base 10 to a form
computers can read, most often an on and an off, 1, 0.
Called Binary,
to compute, ex 26, 11010
from decimal to binary, successive division
so you divide 26 by 2, the remainder becomes the LSB,
the least significant bit. You repeat the process over and
over until you can not divide anymore
from binary to decimal, weighted multiplication
each bit is a power of two more than the previous
so the LSB is 2^0, and then 2^1, 2^2, etc.
Uses, Calendar days. Marginally useful
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there are two algorithms that use Mod 7 to calculate the
day of the week.
One is designed for computers, the other is designed for
mental calculations
the computer algorithm is called the Zeller's Congruence
the mental algorithim is the Doomsday Rule,no relation
to the end of the World...
the Doomsday Rule, this is somewhat
useful
Determine the "anchor day" for the century.
Use the anchor day for the century to calculate the
doomsday for the year.
3. Choose the closest date out of the ones that always fall
on the doomsday (e.g. 4/4, 6/6, 8/8), and count the
number of days (mod 7) between that date and the date
in question to arrive at the day of the week.
Some selected year's doomsday's:
2004 Sunday
2009 Saturday
2005 Monday
2010 Sunday
2006 Tuesday
2011 Monday
2007 Wednesday 2012 Wednesday
2008 Friday
2013 Thursday
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2.
Examples
Find the day of the week on 8/11 when the
doomsday is Wednesday
Find the day of the week on 2/11 when the
doomsday is Friday
Uses, The Musical Scale
Mod 12 is used in the Music World.
In the chromatic scale (one that plays every note between
octaves), there are twelve notes per octave.
The layout of a piano is very convenient for demonstrating
this.