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Re: Root (EXACT Square Root Algorithm)
Posted:
Jul 12, 1999 6:24 PM


This method for square roots is EXACT for the number of digits that you have the patience to use it to.
To view this file properly, convert it/view it with a 'nonproportional' font (eg Courier New in Windows).
This method was formerly taught in primary school, but the algorithm seems to be 'getting lost' today!!!  
To find the square root of any number to any number of digits: ****************************************************************************
(This algorithm looks like long division, but a bit more tedious)
Write down the number:
1234567890.
Starting at the decimal point and working to the left, separate the number into groups of two:
12 34 56 78 90.
leave lots of room to the left: 12 34 56 78 90
Add some notation (pretend these added lines are solid): ________________  12 34 56 78 90
1. What is the largest number squared that will subtract from the first group? (It is obviously 3 here) Square this number and perform the subtraction: 3 ________________  12 34 56 78 90 9 ___ 3 (This sequence of digits above the bar will be referred to as the 'top row'.)
2. Drop down the next group of two digits:
3 ________________  12 34 56 78 90  9  ___  3 34
3. Double the top row and bring this doubled value down to the left, (we'll call this the left row) leaving a placeholder for a less significant digit:
3 ________________  12 34 56 78 90  9  ___ 6_ 3 34
4. Now, we need to find the one digit such that the product of this digit and the current left row with this digit appended is just less than the result of the last subtraction. Put this digit in the placehoder and append it to the top row. ( i.e. in this particular case, the digit is 5, since 5*65<334, but 6*66>334) Perform this multiplication and subtraction:
So we now have: 35 ________________  12 34 56 78 90  9  ___ 65 3 34 3 25 ____ 9 5. Loop starting at step 2 until you have achieved the desired precision.
(continuing the process so you can see the result develop:)
35 ________________  12 34 56 78 90  9  ___ 65 3 34  3 25  ____ Step 2: 9 56
35 ________________  12 34 56 78 90  9  ___ 65 3 34  3 25  ____ Step 3: 70_ 9 56
Step 4: Next digit is 1 since 1*701<956, but 2*702>956)
351 ________________  12 34 56 78 90  9  ___ 65 3 34  3 25  ____ 701 9 56  7 01  ____  2 55
continuing on:
351 ________________  12 34 56 78 90  9  ___ 65 3 34  3 25  ____ 701 9 56  7 01  ____ 702_ 25578
3513 ________________  12 34 56 78 90  9  ___ 65 3 34  3 25  ____ 701 9 56  7 01  ____ 7023 25578 21069 _____ 4509
3513 ________________  12 34 56 78 90  9  ___ 65 3 34  3 25  ____ 701 9 56  7 01  ____ 7023 25578  21069  _____ 7026_ 450990
35136 ________________  12 34 56 78 90  9  ___ 65 3 34  3 25  ____ 701 9 56  7 01  ____ 7023 25578  21069  _____ 70266 450990 421596 ______ 29394.00
so, at this point, to an approximation, sqrt(1234567890) is 35136.
You are only limited by your patience as to how many digits you require.
Thomas Chrapkiewicz 06ap98
 
User wrote in message ... > >Does anyone know how to go about doing a square root or cube root >calculation? > >If I gave you a number (no calculators), and a blank piece of paper, >can you describe the steps one by one how you go about calculating >the cube or square root? (or any root for that matter). > >I also assume I can say... I want it to a precision of 80 decimal points. >You would go about using your method and you would come up with >80 decimal points (or less if it rounds off perfectly). > >Where can I find procedures on doing this? Is there a very simply >way to do this? using basic operators (no log sin cos, etc). > >Thank you. > >Thank you. > > >



