Difference between revisions of "CSC103: DT's Notes 1"
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::''Suppose a lily is in the middle of a lake, and every day its size is twice its size the previous day. In 30 days the lily has covered half of the lake. How long will it take it to cover the whole lake?'' | ::''Suppose a lily is in the middle of a lake, and every day its size is twice its size the previous day. In 30 days the lily has covered half of the lake. How long will it take it to cover the whole lake?'' | ||
| − | 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! | + | 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 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. One is the story of the man who did a good deed and when asked what he wanted for his reward asked for grains of wheat 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 is 2<sup>64</sup>, or 18,446,744,073,709,551,616, much more than the quantity of wheat produced on earth in a single year. | ||
<br /> | <br /> | ||
| + | Let's stay with this second example and write down the number of the square followed by the quantity of grains: | ||
| + | |||
| + | <center> | ||
| + | {| class="wikitable" style="text-align: center; color: green;" | ||
| + | ! square | ||
| + | ! grains | ||
| + | |- | ||
| + | | 1 | ||
| + | | 1 | ||
| + | |- | ||
| + | | 2 | ||
| + | | 2 | ||
| + | |- | ||
| + | | 3 | ||
| + | | 4 | ||
| + | |- | ||
| + | | 4 | ||
| + | | 8 | ||
| + | |- | ||
| + | | 5 | ||
| + | | 16 | ||
| + | |- | ||
| + | | 6 | ||
| + | | 32 | ||
| + | |- | ||
| + | | ... | ||
| + | | ... | ||
| + | |} | ||
| + | </center> | ||
| + | The left column represents the number of the square being covered, and this number increases by 1 every time. The second column represents the quantity of interest, the number of grains, and doubles every time. So that is the setup for studying our exponential growth: something that doubles in size every fixed interval, in our case every new square. | ||
| + | |||
| + | Let's plot these numbers to see their growth. | ||
| + | <br /> | ||
| + | <center>[[Image:CSC103_ExponentialGrowth1.png|300px]]</center> | ||
| + | <br /> | ||
| + | Note the quick growth of the circles. But to get the full impact of the exponential growth we need to plot a few more points, as illustrated in the plots below: | ||
| + | <br /> | ||
| + | <center> | ||
| + | [[Image:CSC103_ExponentialGrowth2.png|200px]] | ||
| + | [[Image:CSC103_ExponentialGrowth3.png|200px]] | ||
| + | [[Image:CSC103_ExponentialGrowth4.png|200px]] | ||
| + | </center> | ||
Revision as of 15:50, 29 September 2013
--© D. Thiebaut 08:10, 30 January 2012 (EST)
