Fractional Graphs and Flowers

denominator

13

base

10

mode

flower

graph

1 / 13	=	0. 076923
2 / 13	=	0. 153846
3 / 13	=	0. 230769
4 / 13	=	0. 307692
5 / 13	=	0. 384615
6 / 13	=	0. 461538
7 / 13	=	0. 538461
8 / 13	=	0. 615384
9 / 13	=	0. 692307
10 / 13	=	0. 769230
11 / 13	=	0. 846153
12 / 13	=	0. 923076

It is common knowledge that every rational number (fraction) can be written as a repeating or terminating decimal. This Demonstration shows a colorful visualization of repeating expansions of rational numbers, highlighting the cycles. Choose the denominator and base with the sliders. Each view shows all (proper) fractions at once. In "flower" mode, first find the red numerator—the numerators are arranged in order counterclockwise around the circle. The first digit of the expansion is the black circled number. Then follow the arrows to find the successive digits. In "graph" mode, disconnected cycles are more clearly separated. (Terminating decimals are shown as a cycle of repeating 0s.)