While finite sequences are specified by listing all elements, this approach does not work for "infinite sequences" which carry on indefinitely. We prefer the more modest term of 'on-sequence', and stress the importance of building up such a theory one step at a time. This lecture begins such a theory by first defining constant on-sequences, which turn out to be finite expressions such as [3) (actually a pointy right bracket) with the possibility of an index to define the m-th element, all of which are 3 in this case. The arithmetic of such constant on-sequences parallels, in an obvious way, the arithmetic of natural numbers.The next step is to define polynumber (or polynomial) on-sequences. We will use the letter n here to characterize such objects, for example the on-sequence [n), enclosed in a square and a pointed bracket, is the basic natural number sequence which we usually would write 1,2,3,.... Again the arithmetic of such polynumber on-sequences parallels that of polynumbers.We give some examples of particular polynumber on-sequences that arise in Sloane's Online Encyclopedia of Integer Sequences (OEIS). In particular we have a look at the frog-and-toad-hopping-interchange sequence :). Is there a convenient way to access all of the polynumber on-sequences in OEIS??We also take this opportunity to dedicate this series to the Australian taxpayers!
Does mathematics make logical sense? No, it does not. Foundational issues have been finessed by modern mathematicians, and this series aims to turn things around. And it will have interesting things to say also about mathematics education---especially at the primary and high school level. The plan is to start right from the beginning, and to define all the really important concepts of basic mathematics without any waffling or appeals to authority. Roughly we discuss first arithmetic, then geometry, then algebra, then analysis, then set theory. This course is aimed for a general audience, interested in mathematics, or willing to learn.