Draft:Square root of 8

Illustration of the square root of 8 as the diagonal of a square with corners at unit coordinates

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 exactly twice the square root of 2.

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

2.82842712474619009760337744841939615713934375075389614635335....

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.

As a periodic continued fraction, the square root of 8 can be represented with a simple repeating pattern of integers:

${\sqrt {8}}$

= [2; 1, 4, 1, 4, ...]

The On-Line Encyclopedia of Integer Sequences lists the sequence for the decimal digits of the square root of 8 at A010466.

The problem of the longest ladder that can be carried around a corner in a hallway of width 1. The illustrated trajectory is not unique except at the tightest spot.

It is the second Lagrange number, and "is the length of the longest (rigid) ladder that can be carried horizontally around a right angled corner in a hallway of unit width".

See also

• Square root of 2
• Square root of 4

References

This open draft remains in progress as of October 15, 2025.