Search results
Results from the WOW.Com Content Network
The principal square root of minus one is i. It has another square root -i. I really dislike the expression "the square root of minus one". Like all non-zero numbers, -1 has two square roots, which we call i and -i. If x is a Real number then x^2 >= 0, so we need to look beyond the Real numbers to find a square root of -1. Complex numbers can be thought of as an extension of Real numbers from ...
-1 is 1 rotated over $\pi$ radians. The square root of a number on the unit circle is the number rotated ...
Also a wanted property is that it is continuous except for the non-positive reals. And I guess it's also a wanted property that for all numbers except of the negative reals, the square root of the conjugate is the conjugate of the square root (for the negative reals, it's not possible to achieve that). $\endgroup$ –
Think about a decimal number between 0 and 1 as a fraction with its numerator GREATER than its denominator. Say you are taking the square root of the number $1/25$. So, you acquire $\sqrt{1/25}$ as the expression which you have to evaluate. This becomes $\sqrt{1}/\sqrt{25}$, or $1/5$. $1/5 > 1/25$.
There is a bit more complicated, but more thorough explanation, however, involving complex analysis. The problem lies in trying to take fractional exponents of negative numbers, e.g. $(-1)^{1/2}$.
And one can quickly check that $(x_3)^2=2.000006007\dots$, which is pretty much the square root of $2$. Share.
In a field such as $\,\mathbb Q,\ \mathbb R,\ \mathbb C,\,$ we have $ \ x^2 = 1 \iff (x-1) (x+1) = 0\iff x = \pm 1.\, $ In rings that are not fields there can be more than two square-roots, e.g. modulo $15$ there are two additional roots $ \ (\pm\,4)^2\equiv 1\pmod{\!15}.\,$ In some contexts authors define single-valued square-root functions that uniformly select one of the roots, e.g. the non ...
When you say "the square root of $49$", you don't really mean what you say: $49$ has two square roots, namely $7$ and $-7$. 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 reason that often only the Taylor series for $\sqrt{1 + x}$ is given in the books is that – for the square-root function – the general case can easily be reduced to the special case: $$ \sqrt {\mathstrut x} = \sqrt {\mathstrut x_0 + x - x_0} = \sqrt {\mathstrut x_0}\sqrt {1 + \frac {\mathstrut x-x_0}{x_0}} $$ and now you can use the ...
I would like to know if there is formula to calculate sum of series of square roots $\sqrt{1} + \sqrt{2}+\dotsb+ \sqrt{n}$ like the one for the series $1 + 2 +\ldots+ n = \frac{n(n+1)}{2}$.