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Topic: aprroximating square roots
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What is the square root of 69, sqrt(69).
69 = 64 + 5
5 is 7.2 percent of 69
3.6% of 8 is .29
sqrt(69) is approximently 8.3
a calculator gives 8.3066
Remember (x+1)*(x+1) = x^2 +2x + 1
consider
(101)*(101) =
(100+1)*(100+1)=
10000 + 200 +1
If we ignore 1 part in 10,0000 then 100 +1% squared is 100^2 +2%. When the % difference is small we can add the percent differences and apply that to the answer. Similairly when we take the square root we can Half tge percent difference.
I will clarify later.
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Please do! I know a couple of tricks, but they are the basic addition and subtraction ones that were useful in the days cashiers still put in your due sum by hand.
The human brain is still the best tool we have. I've known my father to know travel times on foot or on a bicycle up to the exact minute (if you could keep up with his pace!) At a garden I worked at, one of the gardeners once said that it was now four degrees Centigrade. I thought "What?!" and headed straight for the nearest thermometer and sure enough, he was right.
Intuition, approximation and the senses can still be very useful in getting a near-enough answer in the case of abstract data expressed as a numerical value on an arbitrary scale, like temperatures or weights.
But I digress. 3.6 is not sqrt(7.2) but half of 7.2. My computer calculator proves you right, but what's with using half the percentage of what remains above a square root of a whole integer and applying that to the square root figure, then adding it... well, you get it by now.
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