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A Procedure to Compute the Digit Sequence of a Square Root

​
square root of:
7
steps
96
7
10.10100101010011111111010100111010010111110001110100110110111100100
2
r
s = digit sequence of
n
This Demonstration visualizes a procedure for generating the base-2 digit sequence of a square root. From A New Kind of Science, p. 141: "To find
n
, start by setting
r=n
and
s=0
. Then at each step apply the rule
{r,s}→If[r≥s+1,{4(r-s-1),2(s+2)},{4r,2s}]
. The result is that the base-2 digits of
s
turn out to correspond exactly to the digits of
n
. If
n
is not between 1 and 4, it must be multiplied or divided by an appropriate power of 4 before starting the procedure."
The digit sequence of r is shown on the left, while the digit sequence of
s
—which corresponds exactly to
n
—is shown on the right. Black cells represent ones, and gray cells represent zeros.
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