Metric vs. other measurement systems - chart

[mklotz]

...I still know how to find the square root of a number with pencil and paper.

I was taught that awkward system in grade school. I soon purged it from my memory bank. There's a much simpler way to find roots using Newton's method.

Consider this iterative equation...

x2 = (x1 + N / x1) / 2

where:

N = number whose root is to be found
x1 = initial estimate of root
x2 = improved estimate of root

We make a guess of the root, x1, plug it into this equation and get an improved estimate, x2. If this estimate isn't good enough for out purposes, we set x1 = x2 and repeat the process to get an even better estimate. This iterative procedure is repeated until x2 squared is close enough to N to satisfy our requirements.

An example will demonstrate. Let's find the square root of 6 (for reference, my calculator says 2.44949). Six is between 4 (2 squared) and 9 (3 squared) so a reasonable guess for x1 might be 2.5. Then

x2 = (2.5 + 6/2.5)/2 = 2.45 (squared = 6.00250)

Very close but let's do one more iteration...

x2 = (2.45 + 6/2.45)/2 = 2.44949 (squared = 6.00000026)

which would be good enough for most purposes.

I can hear you saying, "Yeah, but you picked an initial guess that was very close to the actual value!" A valid objection so let's try it with an initial guess that's downright silly, 7. Remember, 7 squared is 49, and that's far enough from 6 that even a schoolkid would know it's not a good choice. In addition, the square root of a number can never be greater than the number so, since 7 is greater than 6, choosing it as a first guess is particularly dumb.

x2 = (7 + 6/7)/2 = 3.9286

x2 = (3.9286 +6/3.9286)/2 = 2.7279

x2 = (2.7279 + 6/2.7279)/2 = 2.4637

x2 = (2.4637 + 6/2.4637)/2 = 2.4495

which, squared, is 6.0002..., close enough for the work I do. So, even with a laughable initial guess we got five significant figures in only four iterations. Clearly, one doesn't need to be a genius mathematician to make useful guesses.

Of course, today most four-banger calculators have square root keys. My question is, why ? The average four-banger user wouldn't know a square root from a rutabaga.