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((m1)*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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