NonCalculator Square Roots Up to 1960, and the
Computer Age, if you wanted to find square roots at all accurately, you either
used a mindnumbing ‘guessandcheck’ long multiplication approach – or an
ingenious method attributed to Newton in about 1670. You’re going to use this
last bit of Fraction Wizardry … Step 1
To find the square root of, say, 5, we make a first guess – let’s call it
a_{1} = 2. We know this is an underestimate, but that’s no problem Step 2 Since a square of area 5 has the same area as a rectangle of area 5 (not the genius part !) , we can use a_{1} and b_{1} as ‘lower’ and ‘upper’ estimates for
We deduce the value of
b_{1} from the fact that a_{1}
x b_{1} = 5, so
Step 3 Since
a_{1} is an underestimate, and
b_{1} is an overestimate, we take their average as our
new, improved estimate for
… So, The Problem :A
Your first task is to repeat this procedure ( using
a_{2} instead of a_{1}
) to find an even better estimate of
 I think you will be surprised how
good it is after even just these 2 repetitions. B Then try the same method, from scratch, to find a good estimate of the value of
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