Difference between revisions of "CSC103: DT's Notes 1"
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If you answered 31 days, then congratulations! Indeed, if it doubles in size every day, then after Day 30 it will be twice half the size of the lake, so it will have covered the whole lake! This simple example shows the powerful nature of exponential growth. It took the lily 30 days to cover 50% of the lake, but it takes in only 1 day to cover the other 50%. | If you answered 31 days, then congratulations! Indeed, if it doubles in size every day, then after Day 30 it will be twice half the size of the lake, so it will have covered the whole lake! This simple example shows the powerful nature of exponential growth. It took the lily 30 days to cover 50% of the lake, but it takes in only 1 day to cover the other 50%. | ||
| − | There are many such puzzles in our cultures that attempt to demonstrate the extraordinary power of exponential growth through seemingly impossible feats. Another is the story of the man who did a good deed and when asked what he wanted for his reward asked for grains of | + | There are many such puzzles in our cultures that attempt to demonstrate the extraordinary power of exponential growth through seemingly impossible feats. Another is the story of the man who did a good deed and when asked what he wanted for his reward asked for grains of rice to be placed on a chessboard, one grain on the first square, then 2 grains on the next square, then 4 on the 3rd, 8 on the 4th, and so on. Of course it is impossible to grant such as wish, as the number of grains that would have to be put on the 64th square of the 8-by-8 chessboard is 2<sup>64</sup>, or 18,446,744,073,709,551,616, much more than the quantity of rice produced on earth in a single year. |
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Let's stay with this second example and write down the number of the square followed by the quantity of grains: | Let's stay with this second example and write down the number of the square followed by the quantity of grains: | ||
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| − | So, is there a better way to display the growth of the number of grains of | + | So, is there a better way to display the growth of the number of grains of rice over the complete range of squares of the chessboard, one that would show that there was growth at all levels? The answer is yes. We can use a ''logarithmic'' scale to |
display the quantities of grains. In a logarithmic scale, numbers are arranged in such a way that any pair of numbers that are related to each other in such a way that one is half the size of the other will always be the same distance from each other on the scale. The figure below shows a horizontal logarithmic scale. | display the quantities of grains. In a logarithmic scale, numbers are arranged in such a way that any pair of numbers that are related to each other in such a way that one is half the size of the other will always be the same distance from each other on the scale. The figure below shows a horizontal logarithmic scale. | ||
Revision as of 19:10, 30 September 2013
--© D. Thiebaut 08:10, 30 January 2012 (EST)
