Calculating square roots, cube roots and n th roots of a real number
Calculating square roots
.
We consider the problem of calculating
=
2
1
2
2, that is the number which, when multiplied with itself, gives 2.
2
must lie between 1 and 2, because
=
1
2
1, which is less than 2, while
=
2
2
4, which is greater than 2.
Try squaring the number
3
2
which is midway between 1 and 2.
Since
3
2
2
=
9
4
is greater than 2, we see that
3
2
is greater than
2.
If
r
represents the square root of 2, then
=
r
2
2
and
=
2
r
r
, that is, dividing
2 into 2 gives
2.
More concisely,
=
2
2
2,
because
2
x
=
2
2.
An alternative way to see that that
3
2
is greater than
2 is from the fact that when
3
2
is divided into 2, the resulting number
=
2
3
2
4
3
is
less than
3
2
.
In fact, since
3
2
is greater than
2,
=
2
3
2
4
3
is less than
2, that is
4
3
<
2
<
3
2
.
Since
4
3
<
2
<
3
2
, we could try taking the number midway between
4
3
and
3
2
, namely
+
3
2
4
3
2
=
17
12
,
and check whether
17
12
is
equal to, or close to,
2.
In fact
17
12
is a slightly greater than
2, since
17
12
2
=
289
144
,
which is
2 +
1
144
.
Dividing
17
12
into 2 gives
=
2
17
12
24
17
, which must be slightly less than
2.
Note that
24
17
~
1.411764706
and
17
12
~
1.416666667.
Since
24
17
<
2
<
17
12
, the number midway between
24
17
and
17
12
, namely
577
408
, is likely to be closer to
2
than either
24
17
or
17
12
.
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The number midway between
24
17
and
17
12
is
1
2
+
24
17
17
12
=
1
2
+
24
.
12
17
.
17
204
=
+
288
289
408
=
577
408
.
We can continue this process to calculate
2 as accurately as we want.
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 Spring '08
 Uri
 Square Roots, 12 digits, 10 digits, Peter Stone, 13 digits, 17 g

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