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Activities/Turtle Art/Tutorials/Counting
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Revision as of 23:27, 29 July 2011

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→Discussion: Correction: terameter to petameter

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* scientific notation, such as 1e12 for a trillion/tera.

* scientific notation, such as 1e12 for a trillion/tera.

−A light-year is about 10 trillion kilometers, or 10 ~~terameters~~, or 1e13 m.

+A light-year is about 10 trillion kilometers, or 10 petameters, or 1e13 m.

Going in the other direction, we have milli/1e-3, micro/1e-6, nano/1e-9, pico/1e-12, femto/1e-15. A neutron is about a femtometer across, so escaping neutrinos pass about 1e15 neutrons per meter of travel inside a neutron star. Its density is something like 1e45 neutrons per cubic meter. (At that density, physicists don't know whether they all remain neutrons, or turn into something else, perhaps strange matter containing strange quarks.)

Going in the other direction, we have milli/1e-3, micro/1e-6, nano/1e-9, pico/1e-12, femto/1e-15. A neutron is about a femtometer across, so escaping neutrinos pass about 1e15 neutrons per meter of travel inside a neutron star. Its density is something like 1e45 neutrons per cubic meter. (At that density, physicists don't know whether they all remain neutrons, or turn into something else, perhaps strange matter containing strange quarks.)

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Notice that counting does not actually require numbers. We could count a bag of pebbles against another bag of pebbles, with the result that one has more pebbles than the other, or the reverse, or that the numbers are equal. We can count the people in a theater or stadium that is full to capacity by noticing that there is a one-to-one mapping between people and seats defined by who is sitting in each one.

Notice that counting does not actually require numbers. We could count a bag of pebbles against another bag of pebbles, with the result that one has more pebbles than the other, or the reverse, or that the numbers are equal. We can count the people in a theater or stadium that is full to capacity by noticing that there is a one-to-one mapping between people and seats defined by who is sitting in each one.

−The Italian mathematician Giuseppe Peano did the most successful work on [http://en.wikipedia.org/wiki/Peano_axioms defining counting], or rather the counting numbers, also called the natural numbers. It turns out that there is a hole in his definition that gives us the option of counting beyond infinity, in what is called [http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic Non-Standard Arithmetic]. But we can't do that with ~~a turtle~~, so I'll let it go.

+The Italian mathematician Giuseppe Peano did the most successful work on [http://en.wikipedia.org/wiki/Peano_axioms defining counting], or rather the counting numbers, also called the natural numbers. It turns out that there is a hole in his definition that gives us the option of counting beyond infinity, in what is called [http://en.wikipedia.org/wiki/Non-standard_model_of_arithmetic Non-Standard Arithmetic]. But we can't do that with the Turtle, so I'll let it go for now.

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