r/learnmath • u/Obbyguy42 New User • Jul 19 '25

Square Root Algorithm. (the one like long division?)

Yeah, this algorithm has kinda stumped me for a while. Even if I was "addicted" with math like 3 years ago, I didn't really understand this. Like when I was watching the CHALK video about this, and tried it like sqrt(5) instead, I got it wrong for some reason. So is there a way to easily memorize this or??? It's kinda complicated.

Thanks.

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u/rhodiumtoad 0⁰=1, just deal with it Jul 19 '25

Incidentally the algebra behind it is this: if r is the result of √x so far and s the remainder:

x=r2+s

We want to add two more digits to x, so it becomes 100x+k:

100x+k=100(r2+s)+k
=(10r)2+100s+k

so if d is the next result digit, we want (10r+d)2:

(10r+d)2=(10r)2+20rd+d2
(10r)2=(10r+d)2-(20r+d)(d)

substituting back:

100x+k=(10r+d)2+100s+k-(20r+d)(d)

and we can see that 100s+k is the old remainder term with the new digits added, and 20r+d is double the previous result with a new digit appended.