User:Michael Retriever/Roots
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Newton presented this manipulation of his binomial theorem, stating that it was a much shorter way to extract roots.
[edit] Extraction of roots
- Given the binomial theorem where r is any real number
- we can equally use it for any rational number. For this explanation we'll use r = 1/b
![\sqrt[b]{x+y}=\,](../../../math/1/6/0/16093b8f7c5b8a667e4d81dfab78fd06.png)
- If we apply this to
![\sqrt[b]{1-x}](../../../math/d/6/2/d62bf35366b2f2c8b87e61eb0d1be4aa.png)
we get
![\sqrt[b]{1-x}=1+{\frac{1/b}{1!}}(-x)+{\frac{(1/b)(1/b-1)}{2!}}(-x)^2+{\frac{(1/b)(1/b-1)(1/b-2)}{3!}}(-x)^3+\dots\,](../../../math/7/d/7/7d752948bdbc1649cf522c24806df3b8.png)
- Now let's check the way to extract roots with an example. We know that
- Using the previous infinite expansion with
we get that
- thus
- The true square root of 7 is 2.64575131.... If we make an approximation of the formula with the fractions shown, we get that the square root of 7 is 2.64575289..., which is correct up to the 5th digit after the dot. That is a fairly good approximation taking into account that we didn't need a calculator to do it.
