Shiloh estimated √190
by placing the point on the number line below. Explain how his estimate could be improved.
Answers
190
has no square factors, so
√190
does not simplify.
It can be approximated as:
11097222161
805077112
≈
13.784048752090222
Explanation:
The square root of
190
is the non-negative number
x
such that
x
2
=
190
.
If we factor
190
then we find:
190
=
2
⋅
95
=
2
⋅
5
⋅
19
So
190
has no square factors and as a result is not possible to simplify.
We can use a Newton Raphson type method to find successively better rational approximations to the irrational number
√
190
.
Let our first approximation be
a
0
=
14
, since
14
2
=
196
is quite close.
We can use the following formula to get a better approximation:
a
i
+
1
=
a
2
i
+
n
2
a
i
where
n
=
190
is the number for which we are trying to find the square root.
See: How do you find the square root 28? for a slightly easier way of doing this. For simplicity here, I'll use the classic formula above.
Then:
a
1
=
a
2
0
+
n
2
a
0
=
14
2
+
190
2
⋅
14
=
386
28
=
193
14
≈
13.7857
a
2
=
74489
5404
≈
13.78404885
a
3
=
11097222161
805077112
≈
13.784048752090222