## Modular arithmetics

The clock of numbers

 Number: Modulo:

The construction of ordinary numbers can be seen as a clock with ten hours (from 0 to 9) and as many hands as there is digits in the number. As for a clock, when the unit hand makes a full rotation, it goes back to zero while the decade hand moves to the next digit...

In mathematical langage we say that the numbers are represented modulo 10. This is an arbitrary choice but it seems logical regarding our number of fingers. Nevertheless any number can define a modulo and any number can be expressed in any modulo following the clock construction. For example, computers work in binary language, or modulo 2; they express everything with 0 and 1 characters.

The branch of mathematics that studies the representation of numbers in the infinity of modulos is called modular arithmetics.

A number N can be represented modulo m by: $$N = \sum_i c_i m^i$$

Times tables representation

### Control panel

 Table: Modulo: Line width: Point size*: Colors**:

*Disappear for mod>50
**Disabled for table>7

One interesting application of the modular arithmetic is the geometrical representation of the multiplication tables.

Principle:
Connect by a straight line a number and the result by a multiplication factor and cover up all the possibilities for one modulo integer equally reparted on a circle.

Example: table of 3 modulo 10
(you can enter these numbers in the application above to follow more easily)

• 1x3=3=3 so we connect 1 to 3 by a straight line (1➔3)
• 2x3=6=6 so connect 2➔6
• 3x3=9=9 so connect 3➔9
• 4x3=12=2 so connect 4➔2
• ...

If you activate the colors, the line between 4 and 2 appears in violet meaning that 1 lap has been done. Before 1 lap, the lines are blue and they are red for 2 laps (e.g. 7x3=21=1)

Increasing the modulo increases the number of lines. If you go up to several hundreds, the important number of lines crossings creates the illusion of curves (put linewidth=1 for better results). In that case we can see the appearance of complex patterns. For example, remark the appearance of petals on the borders of the circle; the number of petals is equal to the number of the multiplication table -1.

Play with the application and discover the very interesting properties of the multiplication tables!

Gallery

This application was inspired by Mickael Launay who created a similar application and made a youtube video (in french).Watch here.

Modular arithmetics on the web: khanacademy