Square root of 7

Square root of 7

Rationality	Irrational
Representations
Decimal	2.645751311064590590..._10
Algebraic form	$\sqrt{7}$
Continued fraction	$2+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{4+\ddots }}}}$

The rectangle that bounds an equilateral triangle of side 2, or a regular hexagon of side 1, has size square root of 3 by square root of 4, with a diagonal of square root of 7.

File:Slide rule with square roots of 6 and 7.jpg

A Logarex system Darmstadt slide rule with 7 and 6 on A and B scales, and square roots of 6 and of 7 on C and D scales, which can be read as slightly less than 2.45 and somewhat more than 2.64, respectively

The square root of 7 is the positive real number that, when multiplied by itself, gives the prime number 7.

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

2.64575131106459059050161575363926042571025918308245018036833....

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.

Rational approximations

File:1797 square root of 7.png

Explanation of how to extract the square root of 7 to 7 places and more, from Hawney, 1797

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 and 1852, 3 in 1835, 6 in 1808, and 7 in 1797. An extraction by Newton's method (approximately) was illustrated in 1922, concluding that it is 2.646 "to the nearest thousandth".

Geometry

File:Root rectangles up to 6.png

Root rectangles illustrate a construction of the square root of 7 (the diagonal of the root-6 rectangle).

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.

The minimal enclosing rectangle of an equilateral triangle of edge length 2 has a diagonal of the square root of 7.

Due to the Pythagorean theorem and Legendre's three-square theorem, $\sqrt{7}$ 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). $\sqrt{15}$ is the next smallest such number.

Outside of mathematics

File:Dollar reverse with root 7 rectangle on large inner rectangle.png

Scan of US dollar bill reverse with root 7 rectangle annotation

On the reverse of the current US one-dollar bill, the "large inner box" has a length-to-width ratio of $\sqrt{7}$, and a diagonal of 6.0 inches, to within measurement accuracy.

See also

• Square root
• Square root of 2
• Square root of 3
• Square root of 5
• Square root of 6