As I mentioned earlier I am teaching a programming course at Centennial College.
I decided to use a non standard introduction by starting with doing back of the envelope calculations. In particular using what I refer to as Penzias numbers, to the best of my recollection I found them in a biographical sketch of Penzias. They are somewhat non standard since Google doesn't find any references.
This is a number system using just the digits 0, 1, and 3, and only defining multiply, where 3x3=10. Mathematicians will note that 3 equals root 10, and that this is a minor refinement of order of magnitude calculations.
This worked well, the examples (how much water flows through the Mississippi at the widest point) generated big numbers, which lead into scientific notation and multiplication of exponents, which lead into the binary number system.
Also the examples demonstrated the technique of solving a question by breaking it down into smaller parts, which will be a constant theme in the course.
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