Square root of 7
Positive real number which when multiplied by itself gives 7
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The square root of 7 is the positive real number that, when multiplied by itself, gives the prime number 7.
| Rationality | Irrational |
|---|---|
| Representations | |
| Decimal | 2.645751311064590590... |
| Algebraic form | |
| Continued fraction | |
It is an irrational algebraic number. The first sixty significant digits of its decimal expansion are:
- 2.64575131106459059050161575363926042571025918308245018036833....[1]
which can be rounded up to 2.646 to within about 99.99% accuracy (about 1 part in 10000).
More than a million decimal digits of the square root of seven have been published.[2]
Rational approximations
The extraction of decimal-fraction approximations to square roots by various methods has used the square root of 7 as an example or exercise in textbooks, for hundreds of years. Different numbers of digits after the decimal point are shown: 5 in 1773[3] and 1852,[4] 3 in 1835,[5] 6 in 1808,[6] and 7 in 1797.[7] An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".[8]
Geometry
In plane geometry, the square root of 7 can be constructed via a sequence of dynamic rectangles, that is, as the largest diagonal of those rectangles illustrated here.[9][10][11]
The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7.[12]
Due to the Pythagorean theorem and Legendre's three-square theorem, is the smallest square root of a natural number that cannot be the distance between any two points of a cubic integer lattice (or equivalently, the length of the space diagonal of a rectangular cuboid with integer side lengths). is the next smallest such number.[13]
