find the squareroot of 2000
Answers
Answer:
square root is
not possible as it has odd numbers of zeros at the end .
it isn't a perfect square.
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Answer:
√ 2000 = 20 √ 5 = 20 [ 2 ;¯ 4 ] ≈ 44.7
Step-by-step explanation:
If a , b ≥ 0 then √ a b = √ a √ b
So: √ 2000 = √ 400⋅ 5 =√ 400 ⋅ √ 5 = 20 √ 5
Since 5 = 2 2 + 1 is of the form n 2 + 1 , √ 5 has a simple expansion as a continued fraction:
√ 5 = [ 2 ; ¯ 4 ] = 2 + 1 4 + 1 4 + 1 4 + 1 4 + ...
According to how accurate an approximation we want we can terminate this continued fraction at more or fewer terms.
For example: √ 5 ≈ [ 2 ; 4 , 4 ] = 2 + 1 4+ 1 4 = 2 + 4 17 = 38 17
So: √ 2000 = 20 √ 5 ≈ 20 ⋅ 38 17 ≈ 44.71
Actually: √ 2000 ≈ 44.72135954999579392818
As another way of calculating the successive approximations provided by the continued fraction, consider the sequence:
0 , 1 , 4 , 17 , 72 , 305 , ...
where a 1 = 0 , a 2 = 1 , a i + 2 = a i + 4 a i + 1
This is similar to the Fibonacci sequence, except the rule is a i + 2 = a i + 4 a i + 1
instead of
a i + 2 = ai + a i + 1 .
This is strongly related to the continued fraction:
[ 4 ; ¯ 4 ] = 4 + 1 4 + 1 4 + 1 4+ 1 4 + ...
The ratio between successive terms of the sequence tends to
2 + √ 5 (somewhat faster than the Fibonacci sequence does to 1 2 + √ 5 2 )
For example, we can find an approximation for
√ 5 in: 305 72 − 2 = 161 72 Hence √ 2000 ≈ 20 ⋅ 161 72 = 3220 72 = 44.72