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.