Draft:Square root of 8

The square root of 8 is the positive real number that, when multiplied by itself, gives the natural number 8. It is more precisely called the principal square root of 8, to distinguish it from the negative number with the same property. This number appears in numerous geometric and number-theoretic contexts. It can be denoted in surd form as ${\textstyle {\sqrt {8}}}$ and in exponent form as ${\textstyle 8^{\frac {1}{2}}}$ .

It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:

2.82842712474619009760337744841939615713934375075389614635335....

The square root of 8 is thus exactly twice the square root of 2.^[1]^[2]

A convenient rational approximation for the square root of 8 is ⁠17/6⁠ (≈ 2.8333), accurate to within approximately 0.17%. The rational approximation ⁠82/29⁠ (≈2.8276), has an error of less than 0.03%, and the rational approximation ⁠99/35⁠ (≈2.82857142857...) has an error of 0.000144.

References

^ Martin Gardner, Hexaflexagons and Other Mathematical Diversions (2020), p. 106.
^ James Kyle, Mathematics Unraveled: A New Commonsense Approach (1976), p. 116: "The square root of 8 is particularly interesting, since 8 is the product of 2 × 4. Its square root, presumably, would be the product of the square root of 2 times the square root of 4, or twice the square root of 2".

[1] Martin Gardner, Hexaflexagons and Other Mathematical Diversions (2020), p. 106.

[2] James Kyle, Mathematics Unraveled: A New Commonsense Approach (1976), p. 116: "The square root of 8 is particularly interesting, since 8 is the product of 2 × 4. Its square root, presumably, would be the product of the square root of 2 times the square root of 4, or twice the square root of 2".

[1]

[2]