Instructions for the manual extraction of a square root ======================================================= Prepared by Ian W Halliday, BA Hons, AAIBF Snr, ATMG, CL Enquiries to ian @ halliday.com or ringbark @ livejournal.com Barking ======= Before starting, I would like to introduce a concept which I will call "barking". This is finding a number less than something else by multiplication. It comes up a lot in the extraction of square roots, so I will explain it here. Barking incvolves taking a number, multiplying it by 10 and adding something: "sticking a number on gthe end" in other words. We then multiply the resulting number by the number we stuck on the end. So, for example, we can "bark" 2 by having 21 x 1 = 21, 22 x 2 = 44, 23 x 3 = 69, 24 x 4 = 96 and so on. We bark a number fully when we find the largest number as a result less than something else. For example, compared to 75, 2 barks to 23, because 23 x 3 < 75 but 24 x 4 > 75. Three worked examples ===================== Now that this is sorted out, let's go back to extraction of a square root. Take the number whose squareroot is to be extracted, and split it into pairs of numbers working away from the decimal point. The three examples I will use are 15,876, 67,108,864 and 2. So these will become, for our purposes, 1 58 76 67 10 88 64 2 . 00 00 00 00 00 00 00 First step: first digit ======================= The first step in extracting a square root is to look at the leftmost group of one or two numbers. By inspection, take the square root of that: this will be the first number of your answer. So sqrt(1 58 76) = 1xx.xxx sqrt(67 10 88 64) = 8xxx.xxx sqrt(2) = 1.xxxxxxxx Write that first number up on the top line, like you do in long division. Subtract the square of this number from the first group, also like in long division and you should have something like this: 1 8 1 --------------- --------------- --------------- 15876 67108864 2 1 64 1 - -- - 0 3 1 In long division, you would then bring down the next one digit before you carried on, but in square root extraction, you bring down the next two digits, like this: 1 8 1 --------------- --------------- --------------- 15876 67108864 200 1 64 1 - -- - 058 310 100 Second step: next digits ======================== We are now ready for the first round of barking. DOUBLE the number you have as the result, and then bark it by comparison with the last line you have here. Left example: You have 1 in the answer, which you double to 2. Bark that against 58: 22 x 2 works, 23 x 3 is too big, so the answer is 22, which means that the next number in your answer is a 2. Middle example: You have 8 in the answer, which you double to 16. Bark that against 310: 161 x 1 works, 162 x 2 is too big, so the answer is 161, which means that the next number in your answer is 1. Right example: You have 1 in the answer, which you double to 2. Bark that against 100: 24 x 4 works, 24 x 5 is too big, so the answer is 24, which means that the next number in the answer is 4. Write down the products in the next line, subtract them from what was left, and you'll end up with something like this: 1 2 8 1 1.4 --------------- --------------- --------------- 15876 67108864 200 1 64 1 - -- - 22)058 161) 310 24)100 44 161 96 -- --- -- 14 149 4 We also need to bring down the next two digits, so we end up with this: 1 2 8 1 1.4 --------------- --------------- --------------- 15876 67108864 20000 1 64 1 - -- - 22)058 161) 310 24)100 44 161 96 -- --- -- 1476 14988 400 Now it's time for barking again. This time, we have 12, 81 and 14, which double to 24, 162 and 28. Left example: You have 246 x 6 = exactly 1476. Not surprisingly, the first example works out to something nice! Middle example: 1629 x 9 is still less than 14988, so the next digit in the answer is 9. Right example: 241 x 1 is less than 400, but 242 x 2 is too many, so the next digit in the answer is 1. So We end up with this at the end of this stage: 1 2 6 8 1 9 1.4 1 --------------- --------------- --------------- 15876 67108864 20000 1 64 1 - -- - 22)058 161)310 24)100 44 161 96 -- --- -- 246)1476 1629)14988 281)400 1476 14661 281 ---- ----- --- 0 327 119 = You might be able to spot what happens next: we bring down double the partial result, bark it and subtract the result. Like this: 1 2 6 8 1 9 2 1.4 1 4 --------------- --------------- --------------- 15876 67108864 2000000 1 64 1 - -- - 22)058 161) 310 24)100 44 161 96 -- --- -- 246)1476 1629)14988 281) 400 1476 14661 281 ---- ----- --- 0 16382) 32764 2824)11900 = 32764 11296 ----- ----- 0 604 = Of course, there should be absolutely no surprise that the second example is a good one, but there should be many happy hours ahead for anyone who wishes to extract square roots to arbitrary precision. For sqrt(2), the next few lines go like this: 1.4 1 4 2 1 3 5 6 2 ... ------------------- 2000000000000000000 ... 1 - 24)100 96 -- 281) 400 281 ; 281 x 1 --- 2824)11900 11296 ; 2824 x 4 ----- 28282) 60400 56564 ; 28282 x 2 ----- 282841) 383600 282841 ; 282841 x 1 ------ 2828423)10075900 8485269 ; 2828423 x 3 -------- 28284265) 159063100 141421325 ; 28284265 x 5 --------- 282842706) 1764177500 1697056236 ; 282842706 x 6 ---------- 2828427122)6712126400 5656854244 ; 2828427122 x 2 ---------- At this point, I will use that well known phrase "and so on". As we all know that sqrt(2) is irrational, this will carry on for the foreseeable future and beyond. Any questions about this, requests for clarification, suggestions for improvements in a future version or corrections of typos are welcome and will be credited. This version: copyright Ian W Halliday, 13 August 2004 Version 1.0:23AUG2004:First publication ....+....1....+....2....+....3....+....4....+....5....+....6....+....7....+....8