Square Root of 2 - Properties of The Square Root of Two

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.