October 21, 2008
Have you ever been caught off guard when someone walks up to you and asks, "What's the square root of X?" Me, too. Usually, I can remember approximations for most square roots up through, uh, maybe I won't finish that statement. Anyway, sometimes the numbers are just too damn big or I need more digits after the decimal place than I can comfortably work out. It's times like that when you really need Newton's formula:
b represents the number for which you're seeking the square root and x is your first guess. Wanna see how it all works? Of course you do! Observe:
Let's say that you need the square root of 13 and we want to be within 0.00001 of the actual value. For simplicity, we'll make x the same as b, the number we're taking the square root of.
NewX=7, difference is 6
NewX=4.428571, difference is 2.571429
NewX=3.682028, difference is 0.746544
NewX=3.606345, difference is 0.075682
NewX=3.6055514, difference is 0.000794
NewX=3.605551, difference <0.000001
There you have it: the square root of 13 is approximately 3.605551.
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