This page mentions quite an interesting way to calculate square root of any real number, which is elementary enough so that even senior primary students and junior secondary school students can do it (corresponding to about 5th to 8th year of K-12). It looks strikingly similar to long division notation used by these students (back to the era when calculators are not dominant among them). Too unfortunate that I have no chance to learn it during old days (though I doubt teachers know that as well).
However, the example given in that page is too ‘nice’: 46656, with square root equal to 216. I’ll try to use a non-square number for illustration on how to calculate the digits after decimal point as well: 678.9 . In the following illustration, I will assume reader has read and understood the aforementioned page.
First, group the digits in pairs as mentioned in the page, with decimal point outside of any pair; this means the number is splitted as “6 78 9″, not “67 89″ (the decimal point is between 8 and 9). And find the smallest integer so that its square is not larger than the first digit or first pair of digits. In this case, the result is 2, since 22 ≤ 6; this is written as:
2
┌───────
2 │ 6 78.9
│ 4
├──
│ 2
Next, borrow the next pair of digits downward, similar to what is done in long division:
2
┌───────
2 │ 6 78.9
│ 4
├─────
│ 2 78
The left hand side works differently though; in long division the left hand side never changes, but this time the last digit in left hand side (which is 2) is doubled, then multiplied by 10. We have to decide the last digit later as well:
2 ?
┌───────
2 │ 6 78.9
│ 4
├─────
4? │ 2 78
The question mark is the digit we need to find out; the one on top is also the same digit. That means we need to find the largest digit (in question mark) so that 4? × ? ≤ 278. Here the answer is 6 (46 × 6 = 276):
2 6
┌───────
2 │ 6 78.9
│ 4
├─────
46 │ 2 78
│ 2 76
│ ───────
│ 2 90
Basically, repeat the same process over and over; however, this time some special care must be taken. First, after doubling the last digit on left hand side the number become 52 (46+6):
2 6. ?
┌─────────
2 │ 6 78.9
│ 4
├─────
46 │ 2 78
│ 2 76
│ ───────
52? │ 2 90
And we need to skip this turn, since 52? × ? > 290 for any digit (except 0):
2 6. 0
┌─────────
2 │ 6 78.9
│ 4
├─────
46 │ 2 78
│ 2 76
│ ───────
520 │ 2 90
│ ───────
│ 2 90 00
Now we can reiterate the next step normally:
2 6. 0 5
┌────────────
2 │ 6 78.9
│ 4
├─────
46 │ 2 78
│ 2 76
│ ───────
520 │ 2 90
│ ───────
5205 │ 2 90 00
│ 2 60 25
│ ───────
│ 29 75
One can continue forever like this, though I decide to stop here since the illustration so far should be enough to make some esssential points clear. As a caution, check the answer we get and see if this method really works:
26.05 × 26.05 = 678.6025 < 678.9
26.06 × 26.06 = 679.1236 > 678.9
Next article would be an explanation of how it works.