discrete mathematics study of discontinuous numbers
TRANSCRIPT
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Discrete Mathematics
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Study of discontinuous numbers
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Logic, Set Theory, Combinatorics, Algorithms,
Automata Theory, Graph Theory,
Number Theory, Game Theory, Information
Theory
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RecreationalNumberTheory
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Power of 9s
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9 * 9 = 81
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8 + 1 = 9
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Multiply any number by 9Add the resultant digits
togetheruntil you get one digit
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Always 9e.g.,
4 * 9 = 363 + 6 = 9
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Square Root of Palendromic Numbers
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Square Root of123454321
=11111
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Square Root of1234567654321
=1111111
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Leonardo of Pisa, known as Fibonacci. Series first stated in
1202 book Liber Abaci
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0,1,1,2,3,5,8,13,21,34,55,89. . Each pair of previous numbers equaling the next number of
the Sequence.
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Dividing a number in the sequence into the following
number produces theGolden Ratio
1.62
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Debussy, Stravinsky, Bartókcomposed using
Golden mean (ratio, section).
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Bartók’s Music for Strings, Percussion and Celeste
89
2134
21 13
13 21
55 34
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Importance of number sequences to music.
After all, music is a sequence of numbers.
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Pascal’s Triangle
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• The sum of each row results in increasing powers of 2 (i.e., 1, 2, 4, 8, 16, 32, and so on).
• The 45° diagonals represent various number systems. For example, the first diagonal represents units (1, 1 . . .), the second diagonal, the natural numbers (1, 2, 3, 4 . . .), the third diagonal, the triangular numbers (1, 3, 6, 10 . . .), the fourth diagonal, the tetrahedral numbers (1, 4, 10, 20 . . .), and so on.
• All row numbers—row numbers begin at 0—whose contents are divisible by that row number are successive prime numbers.
• The count of odd numbers in any row always equates to a power of 2.
• The numbers in the shallow diagonals (from 22.5° upper right to lower left) add to produce the Fibonacci series (1, 1, 2, 3, 5, 8, 13 . . .), discussed in chapter 4.
• The powers of 11 beginning with zero produce a compacted Pascal's triangle (e.g., 110 = 1, 111 = 11, 112 = 121, 113 = 1331, 114 = 14641, and so on).
• Compressing Pascal's triangle using modulo 2 (remainders after successive divisions of 2, leading to binary 0s and 1s) reveals the famous Sierpinski gasket, a fractal-like various-sized triangles, as shown in figure 7.2, with the zeros (a) and without the zeros (b), the latter presented to make the graph clearer.
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$1 million prize to createformula for creatingnext primes without
trial and error
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• The sum of each row results in increasing powers of 2 (i.e., 1, 2, 4, 8, 16, 32, and so on).
• The 45° diagonals represent various number systems. For example, the first diagonal represents units (1, 1 . . .), the second diagonal, the natural numbers (1, 2, 3, 4 . . .), the third diagonal, the triangular numbers (1, 3, 6, 10 . . .), the fourth diagonal, the tetrahedral numbers (1, 4, 10, 20 . . .), and so on.
• All row numbers—row numbers begin at 0—whose contents are divisible by that row number are successive prime numbers.
• The count of odd numbers in any row always equates to a power of 2.
• The numbers in the shallow diagonals (from 22.5° upper right to lower left) add to produce the Fibonacci series (1, 1, 2, 3, 5, 8, 13 . . .), discussed in chapter 4.
• The powers of 11 beginning with zero produce a compacted Pascal's triangle (e.g., 110 = 1, 111 = 11, 112
= 121, 113 = 1331, 114 = 14641, and so on).• Compressing Pascal's triangle using modulo 2
(remainders after successive divisions of 2, leading to binary 0s and 1s) reveals the famous Sierpinski gasket, a fractal-like various-sized triangles, as shown in figure 7.2, with the zeros (a) and without the zeros (b), the latter presented to make the graph clearer.
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1111 111111 111 111 1 1 1111111111 111 111 1 1 11111 11111 1 1 111 11 11 1 11 1 1 1 1 1 1 111111111111111111 111 111 1 1 11111 11111 1 1 111 11 11 1 11 1 1 1 1 1 1 111111111 1 11111111 1 1 111 11 11 111 1 1 1 1 1 1 11111 1111 1111 11111 1 1 1 1 1 1 111 11 11 1 1 1 1 1 1 1 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1111111111111111111111111111111111 111 111 1 1 11111 11111 1 1 111 11 11 1 11 1 1 1 1 1 1 111111111 111111111 1 1 111 11 11 111 1 1 1 1 1 1 11111 1111 1 111 11111 1 1 1 1 1 1 111 11 11 1 1 11 1 1 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11111111111111111 11111111111111111 1 1 111 11 1 1 1 11 1 1 1 1 1 1 11111 1111 1111 11111 1 1 1 1 1 1 111 11 11 1 1 11 1 1 11 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 111111111 1 1111111 1 1111111 1 11111111 1 1 1 1 1 1 111 11 11 11 11 11 11 111 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11111 1111 1111 1111 1111 1111 1111 1111
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Magic Squares
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Square Matrixin which
all horizontal ranksall vertical columns
both diagonalsequal same number when
addedtogether
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0-2 7 9 -9
-711 -5 2 4
6-1 13 -10 -3
-6-8 1 8 10
12 5 -11 -4 3
1
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6-1 13 -10 -3
-6-8 1 8 10
12 5 -11 -4 3
0-2 7 9 -9
-711 -5 2 4
12 5 -11 -4 3
0-2 7 9 -9
-711 -5 2 4
6-1 13 -10 -3
-6-8 1 8 10
-711 -5 2 4
6-1 13 -10 -3
-6-8 1 8 10
12 5 -11 -4 3
0-2 7 9 -9
-6-8 1 8 10
12 5 -11 -4 3
0-2 7 9 -9
-711 -5 2 4
6 -1 13 -10 -3
0-2 7 9 -9
-711 -5 2 4
6-1 13 -10 -3
-6-8 1 8 10
12 5 -11 -4 3
1 2
3
4 5
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Musikalisches Würfelspiele
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Number of Possibilitiesof 2 matrixes
is1116
or45,949,729,863,572,161
45 quadrillion
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Let’s hear a couple
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Xn+1 = 1/cosXn2
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(defun cope (n seed) (if (zerop n)() (let ((test (/ 1 (cos (* seed seed))))) (cons (round test) (cope (1- n) test)))))
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? (cope 40 2)(-2 -1 -2 -4 -1 -11 -3 2 -1 10 1 -2 -1
2 -9 -2 1 2 29 1 -7 3 -9 -4 1 2 -2 -1 2 -1 3 1 -2 -1 2 4 1 2 -2 -1)
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Tom Johnson’s
Formulas forString Quartet
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No. 7
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Iannis Xenakis
Metastasis