Square root of 6
positive real number which when multiplied by itself gives 6
The square root of 6 is the positive real number that, when multiplied by itself, gives the natural number 6. It is more precisely called the principal square root of 6, to distinguish it from the negative number with the same property. This number appears in numerous geometric and number-theoretic contexts.
It is an irrational algebraic number. The first thirty significant digits of its decimal expansion are:
which can be rounded up to 2.45 to within about 99.98% accuracy (about 1 part in 4800).
Since 6 is the product of 2 and 3, the square root of 6 is the geometric mean of 2 and 3, and is the product of the square root of 2 and the square root of 3, both of which are irrational algebraic numbers.
NASA has published more than a million decimal digits of the square root of six.
Geometry
In plane geometry, the square root of 6 can be constructed via a sequence of dynamic rectangles, as illustrated here.
In solid geometry, the square root of 6 appears as the longest distances between corners (vertices) of the double cube, as illustrated above. The square roots of all lower natural numbers appear as the distances between other vertex pairs in the double cube (including the vertices of the included two cubes).
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