Non-Calculator Square Roots
Up to 1960, and the
Computer Age, if you wanted to find square roots at all accurately, you either
used a mind-numbing ‘guess-and-check’ 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
a1 = 2.
We know this is an under-estimate, 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 a1 and b1
as ‘lower’ and ‘upper’ estimates for
We deduce the value of
b1 from the fact that a1
x b1 = 5, so 
Step 3
Since
a1 is an under-estimate, and
b1 is an over-estimate, we take their average as our
new, improved estimate for
…
So, 
The Problem :
A
Your first task is to repeat this procedure ( using
a2 instead of a1
) 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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