I have another blog that doesn't suck. Archive:
Comments disabled 
Toph and I were discussing the Pythagorean theorem, and I asked her what the square root of five was. She wasn't sure, but said “two, with a remainder one?” I think this is an insightful answer. She is correctly analogizing the operation with integer division. In division, we say that !!a!! divided by !!b!! has a remainder of !!r!! when $$a = bq+r\qquad(0\le r \lt b).$$ If we require the condition on !!r!!, the solution to the equation is unique. The analogous situation for square roots is that the square root of !!n!! has a remainder of !!r!! when $$n = c^2 + r\qquad(0\le r\lt 2c+1).$$ Again, the condition on !!r!! guarantees a unique solution. The world is full of things that aren't important enough to have their own name. What I find interesting about this is that I think it is important enough to have a name. In connection with square roots we often need discuss the remainder, but we don't call it that. (I was going to dig up evidence of this, then lost interest and filed
the article in the “publish someday, maybe” folder. But then I
remembered:
