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Math Workshop Teacher Mobility EUROCALL Project Udine, 6 November 2008 G. Catanese

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Powers of 2 How many times can a sheet of paper be folded in half, at the most? Does it depend exclusively on the size of the sheet? Try..... The limit, without machines, is 7. The ‘most folded’ thing was a very thin sheet of gold, which was folded 12 times.

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Other… curiosities If we fold our sheet 10, 20, 30, 40 times, what thickness do we get? The equivalent of the height of the Eiffel Tower ? Or the length of Italy? Or more? What will the respective sizes of our sheet have to be? What do you think?…

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Scientific notation Before going on, we need to take a look at a form of notation that is extremely useful for writing very small or very large numbers 10 = 1* 10 1 100 = 1*10 2 0.1 = 1* 10 -1 I can write any number to the nearest power of 10, with the initial one-digit- coefficient, i.e.: 4,200,000 = 4.2* 10 6 0.00085 = 8.5*10 -4

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The distance between the earth and the moon is about 380,000 km... 380,000,000 m = 3.8 * 10 8 m 380,000,000,000 mm = 3.8 * 10 11 mm If we suppose that the thickness of a sheet of paper is 0,1 mm = 1* 10 -1 mm = 1* 10 -4 m the distance between the earth and the moon is 3,800,000,000,000 sheets of paper, one on top of the other... As 2 42 = 4,400,000,000,000 = 4.4 * 10 12 this means that it is enough to fold a paper only 42 times, to get a thickness that is greater than the distance between the earth and the moon...

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Supposing we cut the borders of the paper with each successive fold, in order to limit the very considerable vertical dimension... To arrive at a final dimension of a 1-cm-side square, our initial sheet of paper must have a size of: 1* 2 42 cm 2 = 4.4 * 10 12 cm 2 = 4.4 * 10 8 m 2 = 4.4 * 10 2 km 2 that is to say, an extension of 440 km 2 which is 10 times the extension of the Vatican … or 4 times that of Paris.

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Prof. Honsell’s question Do you think it would be possible for me, simply by asking 20 yes/no questions, to guess any word in the Italian language that one of you might choose? Who wants to bet?…

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A comprehensive dictionary has about 500,000 words, written on about 16,000 pages, i.e. 2 14 pages, with every page listing approximately 64 -- i.e. 2 6 -- words. It would therefore be sufficient for me to ask if the word is in the first half of the dictionary and then to continue halving the number of pages in each successive question. I can identify the page with no more than 14 questions, then go on halving until -- with 6 attempts at the most -- I guess the word.

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Explanations and further considerations How can we calculate 2 32 ? Properties of the powers Exponential function

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Exponential Function (0,1) (1,2) Graph for y = 2 x

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Bibliography F. Honsell, L’algoritmo del parcheggio (Ed. Mondadori, 2007)

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