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C00002 00002 Let's look for the square root of 20, a number which gives 20 when multiplied
C00006 00003 Starting with the first number of a pair, call it A the second number of that
C00007 ENDMK
Cā;
Let's look for the square root of 20, a number which gives 20 when multiplied
by itself. There is a more general question that will give us insight into
the particular one (a typical mathematician's trick): what pairs of numbers
can be multiplied to give 20? Examples are 1x20, 2x10, 2.5x8, 4x5. Given
one such number, say 7, we can find the other number by dividing the first
into 20, getting 20/7 = 2.854. This matches up the numbers in pairs, except,
of course, that the square root of 20 is a wallflower, matched with itself.
These pairs are _nested_; that is, given any two pairs, both numbers of one lie
between the numbers of the other. For example, 2.5 and 8 both lie between 2 and
10. It has to be this way. If one number of a pair were less than 2, and another
were between 2 and 10, the product would be less than 20, so that pair would be
no good.
The picture below illustrates this pairing:
The pair of numbers which are both 20 lies inside all the other pairs.
Given any pair of numbers, say 1 and 20, we can pick any number between them,
say 3, and its mate, 20/3 = 6.67, which will also be between the first pair
of numbers. The first pair contain 20 between them, and the second pair
contain it more closely. In this way, we could keep finding numbers closer
and closer to 20 .
To program this for a computer, how can we arrange to pick a number between
the first pair? Lots of ways, but an easy one is to add the numbers and divide
by 2. Let's try this for several steps, starting with 1 and 20.
First pair: 20 and 1 average 10.5
Second pair: 10.5 and 1.905 average 6.202
Third pair: 6.202 and 3.225 average 4.713
Fourth pair: 4.713 and 4.243 average 4.478
Fifth pair: 4.478 and 4.466 average 4.472
Sixth pair: 4.472 and 4.472
The square root is between each of these pairs of numbers, so it is between
4.472 and 4.472 (actually 4.4722719), and 4.472 is the square root of 20.
So all we had to do was to start with any pair, and keep getting closer pairs
until they were so close that we knew pretty accurately what 20 must be.
Starting with the first number of a pair, call it A; the second number of that
pair is 20/A, and the first number of the next pair is (A + 20/A)/2. So we can
find the first number of each pair by this formula without bothering to get
the other.
How do we know when we are finished? When a pair of mated numbers are very
close together. In that case, both are also very close to the numbers of the
next pair, so we can check that. The larger number in successive pairs keeps
getting smaller until it stops changing; we can use this as a test to stop.