>>9444I was thinking about how to hand-multiply infinite (signed-digit) decimals again. Pic related might be more work than necessary, but it does have the advantage of making it less confusing. It also has the advantage of giving the greatest possible number of digits for a fixed number of digits of the input, which might be helpful if further digits of the numbers you're multiplying are difficult to compute.
Most of the process should be self-explanatory, but some things I should explain. First of all, obviously we can't multiply infinitely many digits. The goal here is a process that if carried on long enough, could give you arbitrarily many digits of the answer, assuming you can get the required number of digits of the two numbers you're multiplying.
Second, as mentioned in
>>9462, I'm using signed digits. Digits can be anything in {-9,-8,-7,-6,-5,-4,-3,
-2,-1,0,1,2,3,4,5,6,7,
8,9}. To represent the negative digits, I draw a line over the digit. Everything is still in base 10, but there are now more ways to represent the same number. For example, early on in the computation, I add 140 + 56 and write the answer as 20(-4) rather than 196. When adding the numbers, I chose to convert the 6 to 1(-4) and carry the 1. I'll explain why I did that in a moment.
When a signed-digit infinite decimal number is truncated to a finite number of digits, the uncertainty in the number is ±1 in the final digit. For example, 1.414 could be the truncation of any number between 1.414[(-9)(-9)(-9),rep
eating] and 1.414[999,repeating] inclusive, which are the same thing as 1.413 and 1.415 respectively. At various stages in the calculation, I've computed the uncertainty in the answer at that point. For example, when multiplying numbers truncated to 1.414*1.414, the result is 2.00(-1)404 with an uncertainty of ±0.002829. The uncertainty can be calculated by adding 1414+1414+1. By considering the uncertainty, we can figure out which digits can be taken as final digits of the answer. At the 1.414*1.414 stage, the digits 2.00 can be taken as final since 2.00(-1)404 ± 0.002829 is guaranteed to be between 2.0(-1) and 2.01 inclusive. In the pic, I've drawn boxes around the digits which have been taken as final.
This should explain why I wrote the answer to 1.4*1.
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