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sqrt6

sqrt6, denoted by √6 or sqrt(6), is the positive real number whose square equals 6. It is approximately 2.449489743.

Arithmetic and algebraic properties: √6 is irrational; its minimal polynomial over the rationals is x^2 − 6.

Representation and factors: √6 can also be written as √2 · √3, reflecting that 6 = 2 · 3.

Continued fraction: The simple continued fraction of √6 is [2; 2, 4, 2, 4, …], a periodic expansion

Occurrence and significance: √6 appears in problems involving the Pythagorean theorem, quadratic fields, and Diophantine equations

See also: square root, irrational number, quadratic irrational, continued fraction, Pell’s equation, quadratic field, algebraic integer.

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Therefore
it
is
a
quadratic
irrational
and
an
algebraic
integer.
The
Galois
conjugate
of
√6
is
−√6.
Since
6
is
square-free,
√6
belongs
to
the
real
quadratic
field
Q(√6),
whose
ring
of
integers
is
Z[√6].
This
shows
that
√6
lies
in
the
compositum
of
the
real
quadratic
fields
Q(√2)
and
Q(√3).
with
period
2.
This
periodicity
is
a
characteristic
feature
of
square
roots
of
non-square
integers.
where
lengths
or
coefficients
reduce
to
the
square
root
of
6.
It
is
often
cited
as
a
standard
example
of
a
quadratic
irrational
and
is
used
to
illustrate
irrationality
proofs,
field
extensions,
and
the
properties
of
continued
fractions.