09-17-2021, 11:00 PM

I saw the thread "Third Order Convergence for Square Roots..."

I thought that I would offer up a process that allows calculations of square roots, similar to how long division is done:

Using Albert Chan's 12345 to calculate the square root:

100 + 10 + 1 + 0.1 + 0.00 + 0.008 + .0000 + .00005 = 111.10805

100 _________| 12345

(200) (*100) = -10000

200 __________2345

210 _________-2100

220 ___________245

221 __________-221

222 ____________24

222.1 _________-22.21

222.2 ___________1.79

222.20 __________0.0000 (222.20*.01=2.222 > 1.779)

222.20 __________1.79

222.208 ________-1.777664

222.216 _________0.012336

222.2160 ________0.000000 (222.216*.0001=.00222216 > .001336

222.2160 ________0.012336

222.21605 _______0.001111080525

222.21610 _______0.0012251975

etc.

Basically, we're keeping track of 2*X on the left, then we add a digit δ to get 2X+δ. Finally we multiple by δ to get 2Xδ+δ^2. We subtract this from the previous quotient to get a new quotient.

Note that for the final quotient = 12345-111.10805^2 = .0012251975 is the remainder left over from squaring 111.10805.

I used the process to find square roots of numbers manually to 25+ digits fairly quickly. I have thought of somehow programming this on the HP-48s, but I have not been able to get past the doubling of the one digit at each step in some quick ML way.

I'm sorry about how this looks, but this forum appears to take out all of the extra spaces I had placed to make it more readable. I used underlines to space things out some, but it doesn't quite look right.

Except for the first digit, the rest of the digits added are in bold (as δ), then multiplied by δ. In the next line the digit added is doubled (or add δ again?).

I thought that I would offer up a process that allows calculations of square roots, similar to how long division is done:

Using Albert Chan's 12345 to calculate the square root:

100 + 10 + 1 + 0.1 + 0.00 + 0.008 + .0000 + .00005 = 111.10805

100 _________| 12345

(200) (*100) = -10000

200 __________2345

210 _________-2100

220 ___________245

221 __________-221

222 ____________24

222.1 _________-22.21

222.2 ___________1.79

222.20 __________0.0000 (222.20*.01=2.222 > 1.779)

222.20 __________1.79

222.208 ________-1.777664

222.216 _________0.012336

222.2160 ________0.000000 (222.216*.0001=.00222216 > .001336

222.2160 ________0.012336

222.21605 _______0.001111080525

222.21610 _______0.0012251975

etc.

Basically, we're keeping track of 2*X on the left, then we add a digit δ to get 2X+δ. Finally we multiple by δ to get 2Xδ+δ^2. We subtract this from the previous quotient to get a new quotient.

Note that for the final quotient = 12345-111.10805^2 = .0012251975 is the remainder left over from squaring 111.10805.

I used the process to find square roots of numbers manually to 25+ digits fairly quickly. I have thought of somehow programming this on the HP-48s, but I have not been able to get past the doubling of the one digit at each step in some quick ML way.

I'm sorry about how this looks, but this forum appears to take out all of the extra spaces I had placed to make it more readable. I used underlines to space things out some, but it doesn't quite look right.

Except for the first digit, the rest of the digits added are in bold (as δ), then multiplied by δ. In the next line the digit added is doubled (or add δ again?).