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The square root of the square root of x is therefore $$\sqrt{\sqrt x} = (\sqrt x)^{1/2} = (x^{1/2})^{1/2} ...
The suaqre root of a (non-negative) real number is non-negative by definition, but is there a similar decision for "the" square root of other (complex) numbers? $\endgroup$ – Wolfgang Kais Commented Jul 8, 2023 at 17:59
As traditional known the square root of any number would have two result, its cube root would have three ...
In fact you can take any two numbers which can be added to get 2 (not nesserly 0.01 but at least you should know the root of one of them So for example $\sqrt{2} = {(1+1)^{1/2}}$ Know all what you need is to expand it using bio theorem and for 2 terms you ll get 1.5
Calculators probably use some form of Newton-Raphson, while arbitrary precision libraries will probably use a much faster algorithm to calculate $\ln$ and Newton-Raphson to compute $\exp$ from inverting $\ln$, and then use $\sqrt[n]{a} = e^\frac{\ln(a)}{n}$. $\endgroup$
Since $\sqrt{2}$ is irrational, is there a way to compute the first 20 digits of it? What I have done so far . I started the first digit decimal of the $\sqrt{2}$ by calculating iteratively so that it would not go to 3 so fast.
For instance, the distance formula in any finite number of dimensions is a positive square root - namely, for two points, the distance apart is the square root of the sum of the squares of the respective differences of coordinates.
What is the fastest algorithm for finding the square root of a number? I created one that can find the square root of "$987654321$" to $16$ decimal places in just $20$ iterations. I've now tried Newton's method as well as my own method (Newtons code as seen below) What is the fastest known algorithm for taking the second root of a number?
These are straightforward to visualize in terms of area (volume) of a square (cube). If you have just a pen and paper, you may "miss" on the multiplication once or twice before you find the correct next digit, but the algorithms themselves are well-defined. Google square root by hand and pick your favorite explanation.
The square root function, like all bona fide functions, is single-valued rather than multi-valued, so if we were tasked with creating our own square root function from scratch we would have to make a choice between the two square roots of every positive number as the value the function takes; if we want to further impose continuity (and ...