How
to find a square root
Finding the square root of a number is the inverse
operation of squaring that number. Remember, the square of a number is that
number times itself.
Square of n = n²
Square of 2 = 2*2 = 2² = 4
The perfect squares are the squares of
the whole numbers.
Perfect squares :
1,4,9,16,25,36,49,64,81,100
What is the number you want to find the square root of? Here's one
we'll use:
46656
First, divide the number to be square-rooted into pairs of digits,
starting at the decimal point. That is, no digit pair should straddle a 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 the
left, somewhat as in long division.
+--- ---- ----
| 4 66
56
Find the largest number whose square is less than or equal to the
leading digit pair. In this case, the leading digit pair is 4; the largest
number whose 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 56
Now square that number, and subtract from the leading digit pair.
2
+--- ---- ----
2 | 4
66 56
|-4
+----
0
Extend the left bracket; multiply the last (and only) digit of the
left-hand number by 2, put it to the left of the difference you just
calculated, and leave an empty decimal place next to it.
2
+--- ---- ----
2 | 4
66 56
|-4
+----
4_ | 0
Then bring down the next digit pair and put it to the right of the
difference.
2
+--- ---- ----
2 | 4
66 56
|-4
+----
4_ | 0
66
Find the largest number to put in this blank decimal place such
that that number, times the number already there plus the decimal place, will
be less than 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
66
Now subtract the product you just found.
2 1
+--- ---- ----
2 | 4
66 56
|-4
+----
41 | 0
66
|- 41
+--------
25
Now, repeat as before: Take the number in the left column (here,
41) and double its last digit (giving you 42). Copy this below in the left
column, and leave a blank space next to it. (Double the last digit with carry:
for example, if you had not 41 but 49, which is 40+9, you should copy down
40+18 which 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 56
Now, 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're done. Otherwise, you can keep finding more decimal places for as long as
you want.
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