Modular Arithmetic: In Motion

Topics Covered (18 minutes 47 seconds):

• How to draw a dynamical portrait of the addition or multiplication by one element modulo n.
• That the portrait for “+1” always looks like one big cycle.
• The general description of portraits for “+2”.
• The general fact that the portrait of a function will consist of cycles if and only if the function is bijective, with proof.
• A brief look at a few multiplicative portraits.

Background needed:

• the definitions of function, injectivity, surjectivity, bijectivity
• the ability to use modular arithmetic in practice (see Modular Arithmetic: User’s Manual.)

Materials and links:

• Tool to compute addition and multiplication tables.
• PDF of small addition and multiplication tables.
• Tool to draw modular dynamics pictures.
• Video Follow-Up Sheet (practice after the video).
• Additive Dynamics Exploration (to be done in groups after the video).