Although doubling is a simple concept, an amazing thing
happens if we iterate it many, many times. (Project
a – Doubling allowances) From the initial
input 1, the first iterate of function (1) is 2. Putting
it back into function (1), we get 4 for the second
iterate, 8 for the third iterate, and so on, as shown in
Table 1. In this way, we obtain the following sequence
of iterates
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2, 4, 8, 16, 32, 64, 128 ….
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(2) |
Clearly, the next members of the sequence are 256(= 2 x 128),
512 (= 2 x 256), etc.
By writing the increasing numbers of sequence (2) in powers of 2,
that is, 21. 22, 23, 24…,
we see that doubling has the effect of raising the power exponent
by 1 at each step in the sequence. With the beginning
number of unity 20(=1), the sequence of 20, 21, 22, 23, 24…
forms the bases of the binary number system by which we count
numbers by only 2 fingers, such as, “on”
=1 or “off”=0, as done by a computer
processor. For us, however, with 10 fingers it is more
natural to use the decimal number system with the bases;
100, 101, 102, 103,
…, instead. We shall encounter sequence (2)
whenever doubling takes place.
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