Wednesday, May 24, 2017

Fun with the Fractional Part function and Integers

Fun with the Fractional Part function and Integers

Let n be a integer and frac(n) be the fractional part function.  For example, frac(28.38) = 0.38.

Alternating Ones and Zeros – Ones Assigned to Odd Numbers

f(n) = 2 * frac(n/2) = n mod 2

n
1
2
3
4
5
6
7
8
9
10
f(n)
1
0
1
0
1
0
1
0
1
0

Alternating Ones and Zeros – Ones Assigned to Even Numbers

f(n) = 2 * frac((n + 1)/2)

n
1
2
3
4
5
6
7
8
9
10
f(n)
0
1
0
1
0
1
0
1
0
1

Alternating Integers – Integers Assigned to Odd Numbers, Zeros to Even

f(n) = (2*n) * frac(n/2)

n
1
2
3
4
5
6
7
8
9
10
f(n)
1
0
3
0
5
0
7
0
9
0

Alternating Integers – Integers Assigned to Even Numbers, Zeroes to Odd

f(n) = (2*n) * frac((n+1)/2)

n
1
2
3
4
5
6
7
8
9
10
f(n)
0
2
0
4
0
6
0
8
0
10

Alternate +1 and -1.  Use of superposition of functions

f(n) = (2) * frac((n+1)/2) + (-2) * frac(n/2)

n
1
2
3
4
5
6
7
8
9
10
f(n)
-1
1
-1
1
-1
1
-1
1
-1
1

The Number of Petals of a Rose Function

Odd + Even = (2*n)*frac(n/2) + (4*n)*frac((n+1)/2)

This determines the number of pedals a rose gets.  The rose is represented by the polar equation r = cos(n*θ).  If n is odd, the rose has n petals.  If n is even, the rose has 2*n petals.

n
1
2
3
4
5
6
7
8
9
10
f(n)
1
4
3
8
5
12
7
16
9
20

Other Examples

f(n) = 3 * frac(n/3) = n mod 3

n
1
2
3
4
5
6
7
8
9
10
f(n)
1
2
0
1
2
0
1
2
0
1

f(n) = 3 * n * frac(n/3)

n
1
2
3
4
5
6
7
8
9
10
f(n)
1
2
0
4
10
0
7
14
0
10

f(n) = 3 * n * frac((n+1)/3)

n
1
2
3
4
5
6
7
8
9
10
f(n)
2
0
3
8
0
6
14
0
9
20

f(n) = 4 * frac(n/4) = n mod 4

n
1
2
3
4
5
6
7
8
9
10
f(n)
1
2
3
0
1
2
3
0
1
2

f(n) = 4 * frac(2*n/4) = 4 * frac(n/2)

n
1
2
3
4
5
6
7
8
9
10
f(n)
2
0
6
0
10
0
14
0
18
0

f(n) = 4 * frac(3*n/4) (sort of “reversing” n mod 4)

n
1
2
3
4
5
6
7
8
9
10
f(n)
3
2
1
0
3
2
1
0
3
2

Generalizing:

Let m and n be positive integers.  Hence, m * frac(n/m)  = n mod m

To reverse the sequence, use m * frac((m-1)*n/n)


Don’t forget to play sometimes.  Math can be really fun when you let go. 

Eddie



This blog is property of Edward Shore, 2017.

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