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There are only two square roots of ii (as there are two square roots of any non-zero complex number), namely ± (1 + i) / √2. In the context of your answer, what happens is that the different values are e (πi / 2 + 2πik) / 2 = eπi / 4 + πik; but the value of this depends only on the parity of k, and so gives just two values, namely ± ...
So for example √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. 1 + (0.5) ⋅ 1 = 1.5 Which is not too bad approximation for √2 = 1.4142135624. I have edited your answer to use MathJax, but it is still in need of improvement.
As I said in a comment. ∑i=1N i√ = H(−1 2) N ∑ i = 1 N i = H N (− 1 2) where H H represents a generalized harmonic number. For large values of N N, the following approximation is quite interesting and extremely accurate. ∑i=1N i√ = 2N3/2 3 + N−−√ 2 + ζ(−1 2) + 1 N−−√ 24 + O((1 N)2) ∑ i = 1 N i = 2 N 3 / 2 3 + N 2 ...
Guess what the square root of the irrational number is. For example, if your irrational number is 2, you might guess 1.2. Divide the initial irrational number by the guessed number. For example, 2 divided by 1.2 is 1.67. Add the resulting sum to the original guessed number. For example, 1.67 plus 1.2 is 2.87.
Finding roots of quadratic polynomials, negative square root. 2. Solving for unknown inside square root. 9.
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 ...
You could say "the square roots of $49$ are $\pm 7$" and that would be fine; but otherwise saying "the square root of $49$" usually refers to what we write as $\sqrt{49}$. The $\sqrt{\ }$ symbol always refers to the positive root by default, so although $\sqrt{49}=7$ (which is positive) is 'the square root of $49$', $-\sqrt{49}=-7$ is another ...
$\begingroup$ Minor point: I notice quite a few elementary algebra books as well as some writers here taking the view that the n-th root of x is defined as x to the power 1/n.
For example, for p = 10 p = 10, the exact value is ≈ 22.46827819 ≈ 22.46827819 while the above approximation gives ≈ 22.46827983 ≈ 22.46827983. By itself, the first term already gives 21.0819 21.0819; the sum of first and second term gives ≈ 22.6629 ≈ 22.6629. For p = 100 p = 100, the approximation leads to 12 12 exact significant ...
One way is to convert the complex number into polar form. For z = reiθ, z2 = r2ei (2θ). So to take the square root, you'll find z1 / 2 = ± √reiθ / 2. Added: Just as with the nonnegative real numbers, there are two complex numbers whose square will be z. So there are two square roots (except when z = 0). Share.