Properties of The Square Root of Two
One-half of, also 1 divided by the square root of 2, approximately 0.70710 67811 86548, 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
This number satisfies
One interesting property of the square root of 2 is as follows:
since This is related to the property of silver means.
The square root of 2 can also be expressed in terms of the copies of the imaginary unit i using only the square root and arithmetic operations:
The square root of 2 is also the only real number other than 1 whose infinite tetrate is equal to its square.
The square root of 2 appears in Viète's formula for π:
for m square roots and only one minus sign.
Similar in appearance but with a finite number of terms, the square root of 2 appears in various trigonometric constants:
It is not known whether 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.
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—Ralph Waldo Emerson (1803–1882)
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