Elementary Arithmetic

It is easy not to notice that counting and arithmetic do not require numbers. People have used beads on an abacus, pebbles on a Roman counting board, fingers (and sometimes toes), score marks on a stick, and other devices for counting and simple arithmetic. The earliest numerals were sets of marks in the appropriate number, such as Chinese 一二三. The abacus is capable of quite advanced arithmetic in the hands of a practiced user.

Mathematicians sometimes use purely mathematical constructions such as sets of sets to represent numbers, and abstract axioms to define arithmetic. The possibilities are infinite, but three definitions stand out.

• The Natural Numbers defined by the Peano Postulates, with Gödel-style recursive functions for arithmetic
• The Hyperreal numbers, based on extensions to the Peano Natural Numbers in Non-Standard Arithmetic (Löwenheim, Skolem, Robinson)
• The Surreal numbers and their extension to games of perfect information (John Horton Conway)

(At a comparable point in the song Lobachevsky, Tom Lehrer said, “Bozhe moi! This, I know from nothing!”) Of these three, conventional programming languages such as Python support only the familiar Natural Numbers, and that is all that Turtle Art is set up to use, so we will stick with them for now. But there is software to work with other kinds of numbers, and they are quite useful, so I will come back to them sometime.

• Comparing numbers (< = >)
• Addition and subtraction with lines of dots
• Multiplication with rectangles of dots
• Division with rectangles, and what to do with the dots left over
• Multiplication as repeated addition
• Multiplication defined recursively
• Abacus arithmetic
• Numerals
• Fractions
• Complex numbers

Note

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