In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, a nested radical is a
radical expression
In mathematics, a radicand, also known as an nth root, of a number ''x'' is a number ''r'' which, when raised to the power ''n'', yields ''x'':
:r^n = x,
where ''n'' is a positive integer, sometimes called the ''degree'' of the root. A roo ...
(one containing a square root sign, cube root sign, etc.) that contains (nests) another radical expression. Examples include
:
which arises in discussing the
regular pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
, and more complicated ones such as
:
Denesting
Some nested radicals can be rewritten in a form that is not nested. For example,
Rewriting a nested radical in this way is called denesting. This is not always possible, and, even when possible, it is often difficult.
Two nested square roots
In the case of two nested square roots, the following theorem completely solves the problem of denesting.
If and are
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rat ...
s and is not the square of a rational number, there are two rational numbers and such that
:
if and only if
is the square of a rational number .
If the nested radical is real, and are the two numbers
:
and
where
is a rational number.
In particular, if and are integers, then and are integers.
This result includes denestings of the form
:
as may always be written
and at least one of the terms must be positive (because the left-hand side of the equation is positive).
A more general denesting formula could have the form
:
However,
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
implies that either the left-hand side belongs to
or it must be obtained by changing the sign of either
or both. In the first case, this means that one can take and
In the second case,
and another coefficient must be zero. If
one may rename as for getting
Proceeding similarly if
it results that one can suppose
This shows that the apparently more general denesting can always be reduced to the above one.
''Proof'': By squaring, the equation
:
is equivalent with
:
and, in the case of a minus in the right-hand side,
:,
(square roots are nonnegative by definition of the notation). As the inequality may always be satisfied by possibly exchanging and , solving the first equation in and is equivalent with solving
:
This equality implies that
belongs to the
quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 a ...
In this field every element may be uniquely written
with
and
being rational numbers. This implies that
is not rational (otherwise the right-hand side of the equation would be rational; but the left-hand side is irrational). As and must be rational, the square of
must be rational. This implies that
in the expression of
as
Thus
:
for some rational number
The uniqueness of the decomposition over and
implies thus that the considered equation is equivalent with
:
It follows by
Vieta's formulas
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").
Basic formulas ...
that and must be roots of the
quadratic equation
In algebra, a quadratic equation () is any equation that can be rearranged in standard form as
ax^2 + bx + c = 0\,,
where represents an unknown value, and , , and represent known numbers, where . (If and then the equation is linear, not q ...
:
its
(≠0, otherwise would be the square of ), hence and must be
:
and
Thus and are rational if and only if
is a rational number.
For explicitly choosing the various signs, one must consider only positive real square roots, and thus assuming . The equation
shows that . Thus, if the nested radical is real, and if denesting is possible, then . Then, the solution writes
:
Some identities of Ramanujan
Srinivasa Ramanujan demonstrated a number of curious identities involving nested radicals. Among them are the following:
and
Landau's algorithm
In 1989
Susan Landau introduced the first
algorithm
In mathematics and computer science, an algorithm () is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing ...
for deciding which nested radicals can be denested. Earlier algorithms worked in some cases but not others. Landau's algorithm involves complex
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
and runs in
exponential time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by t ...
with respect to the depth of the nested radical.
In trigonometry
In
trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies ...
, the
sines and cosines of many angles can be expressed in terms of nested radicals. For example,
: