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| 28 May 2016 06:01 PM |
o wsh rthat oroblosxwdipul;d mauhlelekwe gi ldlkfl v beuvhemtes
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iiFerexes
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| Joined: 17 Jul 2014 |
| Total Posts: 8664 |
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| 28 May 2016 06:07 PM |
>ultimate unscramble One-half of √2, also the reciprocal of √2, approximately 0.707106781186548, is a common quantity in geometry and trigonometry because the unit vector that makes a 45° angle with the axes in a plane has the coordinates
\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right). This number satisfies
\tfrac{1}{2}\sqrt{2} = \sqrt{\tfrac{1}{2}} = \frac{1}{\sqrt{2}} = \cos 45^{\circ} = \sin 45^{\circ}. One interesting property of √2 is as follows:
\!\ {1 \over {\sqrt{2} - 1}} = \sqrt{2} + 1 since
\left(\sqrt{2}+1\right)\left(\sqrt{2}-1\right)=2-1=1. This is related to the property of silver ratios.
√2 can also be expressed in terms of the copies of the imaginary unit i using only the square root and arithmetic operations:
\frac{\sqrt{i}+i \sqrt{i}}{i}\text{ and }\frac{\sqrt{-i}-i \sqrt{-i}}{-i} if the square root symbol is interpreted suitably for the complex numbers i and −i.
√2 is also the only real number other than 1 whose infinite tetrate (i.e., infinite exponential tower) is equal to its square. In other words: if for c > 1 we define x1 = c and xn+1 = cxn for n > 1, we will call the limit of xn as n → ∞ (if this limit exists) f(c). Then √2 is the only number c > 1 for which f(c) = c2. Or symbolically:
\sqrt{2}^ {(\sqrt{2}^ {(\sqrt{2}^ {(\ \cdot^ {\cdot^ \cdot)))}}}} = 2. √2 appears in Viète's formula for π:
2^m\sqrt{2-\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}} \to \pi\text{ as }m \to \infty\, for m square roots and only one minus sign.[19]
Similar in appearance but with a finite number of terms, √2 appears in various trigonometric constants:[20]
\sin(\pi/32) = \tfrac12\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2}}}}; \sin(\pi/16) = \tfrac12\sqrt{2-\sqrt{2+\sqrt{2}}}; \sin(3\pi/32) = \tfrac12\sqrt{2-\sqrt{2+\sqrt{2-\sqrt{2}}}}; \sin(\pi/8) = \tfrac12\sqrt{2-\sqrt{2}}; \sin(5\pi/32) = \tfrac12\sqrt{2-\sqrt{2-\sqrt{2-\sqrt{2}}}}; \sin(3\pi/16) = \tfrac12\sqrt{2-\sqrt{2-\sqrt{2}}}; \sin(7\pi/32) = \tfrac12\sqrt{2-\sqrt{2-\sqrt{2+\sqrt{2}}}}; \sin(\pi/4) = \tfrac12\sqrt{2}; \sin(9\pi/32) = \tfrac12\sqrt{2+\sqrt{2-\sqrt{2+\sqrt{2}}}}; \sin(5\pi/16) = \tfrac12\sqrt{2+\sqrt{2-\sqrt{2}}}; \sin(11\pi/32) = \tfrac12\sqrt{2+\sqrt{2-\sqrt{2-\sqrt{2}}}}; \sin(3\pi/8) = \tfrac12\sqrt{2+\sqrt{2}}; \sin(13\pi/32) = \tfrac12\sqrt{2+\sqrt{2+\sqrt{2-\sqrt{2}}}}; \sin(7\pi/16) = \tfrac12\sqrt{2+\sqrt{2+\sqrt{2}}}; \sin(15\pi/32) = \tfrac12\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}. It is not known whether √2 is a normal number, a stronger property than irrationality, but statistical analyses of its binary expansion are consistent with the hypothesis that it is normal to base two.[21]
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brpal
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| Joined: 23 Jun 2011 |
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