Approximations of square roots (1 Viewer)

[RealiseNothing]

If:



Then:



I've brought this up a long time ago, but I just remembered it so I thought I'd make a thread to see if this actually leads to anything more.

Background info: basically I was messing around with square roots last year and found this really cool approximation method, that is, the above.

I'll just do an example, say we wanted to find the square root of 9900:

We will always take such that they are perfect squares on either side of our number we are square rooting. So:



Noting that

Substituting this into our formula gives:



Comparing this to the actual value:



So you can see that this method was off the actual value.

So just wanted to put this out there to see if anyone can make something out of it, and just for general discussion I guess. I find this shit cool.

 

[seanieg89]

 

[jyu]

It works due to local linearity of root functions at large x.