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About site: http://math.arizona.edu/~kerl/doc/square-root.html

Title: Math/Recreations - How To Manually Find A Square Root The way to do find a square root without a calculator.

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How to manually find a square root

How to manually find a square rootHere is an almost-forgotten art: one that, with the advent of electroniccalculators, will likely survive to the twenty-first century only on paper andin the memories of oldsters.What is the number you want to find the square root of?Here's one we'll use:46656First, divide the number to be square-rooted into pairs of digits,starting at the decimal point. That is, no digit pair should straddlea decimal point. (For example, split 1225 into "12 25" rather than"1 22 5"; 6.5536 into "6. 55 36" rather than"6.5 53 6".)Then you can put some lines over each digit pair, and a bar to theleft, somewhat as in long division. +--- ---- ---- | 4 66 56 Find the largest number whose square is less than or equal to the leadingdigit pair. In this case, the leading digit pair is 4; the largest numberwhose square is less than or equal to 4 is 2.Put that number on the left side, and above the first digit pair. 2 +--- ---- ---- 2 | 4 66 56Now square that number, and subtract from the leading digit pair. 2 +--- ---- ---- 2 | 4 66 56 |-4 +---- 0Extend the left bracket; multiply the last (and only) digit of the left-handnumber by 2, put it to the left of the difference you just calculated, andleave an empty decimal place next to it. 2 +--- ---- ---- 2 | 4 66 56 |-4 +---- 4_ | 0Then bring down the next digit pair and put it to the rightof the difference. 2 +--- ---- ---- 2 | 4 66 56 |-4 +---- 4_ | 0 66Find the largest number to put in this blank decimal place such that thatnumber, times the number already there plus the decimal place, will be lessthan the current difference. For example, see if 1 * 41 is ≤ 66, then 2*42≤ 66, etc. In this case it's a 1. Put this number in the blank you left,and in the next decimal place on the result row on the top. 2 1 +--- ---- ---- 2 | 4 66 56 |-4 +---- 41 | 0 66Now subtract the product you just found. 2 1 +--- ---- ---- 2 | 4 66 56 |-4 +---- 41 | 0 66 |- 41 +-------- 25Now, repeat as before: Take the number in the left column (here, 41) anddouble its last digit (giving you 42). Copy this below in the left column, andleave a blank space next to it. (Double the last digit with carry: forexample, if you had not 41 but 49, which is 40+9, you should copy down 40+18which is 58.) Also, bring down the next digit pair on the right. 2 1 +--- ---- ---- 2 | 4 66 56 |-4 +---- 41 | 0 66 |- 41 +--------42_ 25 56Now, find the largest digit (call it #) such that 42# * # ≤ 2556. Here,it turns out that 426 * 6 = 2556 exactly. 2 1 6 +--- ---- ---- 2 | 4 66 56 |-4 +---- 41 | 0 66 |- 41 +--------426 | 25 56 |- 25 56 +------------- 0 When the difference is zero, you have an exact square root and you'redone. Otherwise, you can keep finding more decimal places for as longas you want.Here is another example, with less annotation. 7 . 2 8 0 1 ... +----------------------7 | 53 . 00 00 00 00 00 | 49 +----------------------142 | 4 00 | 2 84 +----------------------1448 | 1 16 00 | 1 15 84 +----------------------14560 | 16 00 | 0 +----------------------145601 | 16 00 00 | 14 56 01 +---------------------- | 1 43 99 00 ...John Kerljohn dot r dot kerl at lmco dot comJuly, 1998Current address (as of 2005):kerl at math dot arizona dot edu

The	way	to	do
find	a	square	root	without
a	calculator.

http://math.arizona.edu/~kerl/doc/square-root.html

How To Manually Find A Square Root 2008 December

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The way to do find a square root without a calculator.

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