Nested radical


In algebra, a nested radical is a radical expression that contains another radical expression. Examples include
which arises in discussing the regular pentagon, 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 numbers 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
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.
A more general denesting formula could have the form
However, Galois theory 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,
. 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 this field every element may be uniquely written with and being rational numbers. This implies that is not rational. 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 that and must be roots of the quadratic equation
its , hence and must be
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 two possible cases for the sign of may be considered simultaneously, by assuming and writing the solution

Some identities of Ramanujan

demonstrated a number of curious identities involving nested radicals. Among them are the following:
Other odd-looking radicals inspired by Ramanujan include:

Landau's algorithm

In 1989 Susan Landau introduced the first algorithm for deciding which nested radicals can be denested. Earlier algorithms worked in some cases but not others.

In trigonometry

In trigonometry, the sines and cosines of many angles can be expressed in terms of nested radicals. For example,
and
The last equality results directly from the results of.

In the solution of the cubic equation

Nested radicals appear in the algebraic solution of the cubic equation. Any cubic equation can be written in simplified form without a quadratic term, as
whose general solution for one of the roots is
In the case in which the cubic has only one real root, the real root is given by this expression with the radicands of the cube roots being real and with the cube roots being the real cube roots. In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one. The nested radicals in this solution cannot in general be simplified unless the cubic equation has at least one rational solution. Indeed, if the cubic has three irrational but real solutions, we have the casus irreducibilis, in which all three real solutions are written in terms of cube roots of complex numbers. On the other hand, consider the equation
which has the rational solutions 1, 2, and −3. The general solution formula given above gives the solutions
For any given choice of cube root and its conjugate, this contains nested radicals involving complex numbers, yet it is reducible to one of the solutions 1, 2, or –3.

Infinitely nested radicals

Square roots

Under certain conditions infinitely nested square roots such as
represent rational numbers. This rational number can be found by realizing that x also appears under the radical sign, which gives the equation
If we solve this equation, we find that x = 2. This approach can also be used to show that generally, if n > 0, then
and is the positive root of the equation x2 − x − n = 0. For n = 1, this root is the golden ratio φ, approximately equal to 1.618. The same procedure also works to obtain, if n > 1,
which is the positive root of the equation x2 + x − n = 0.

Ramanujan's infinite radicals

Ramanujan posed the following problem to the Journal of Indian Mathematical Society:
This can be solved by noting a more general formulation:
Setting this to F and squaring both sides gives us
which can be simplified to
It can then be shown that
So, setting a = 0, n = 1, and x = 2, we have
Ramanujan stated the following infinite radical denesting in his lost notebook:
The repeating pattern of the signs is

Viète's expression for

for pi|, the ratio of a circle's circumference to its diameter, is

Cube roots

In certain cases, infinitely nested cube roots such as
can represent rational numbers as well. Again, by realizing that the whole expression appears inside itself, we are left with the equation
If we solve this equation, we find that x = 2. More generally, we find that
is the positive real root of the equation x3 − x − n = 0 for all n > 0. For n = 1, this root is the plastic number ρ, approximately equal to 1.3247.
The same procedure also works to get
as the real root of the equation x3 + x − n = 0 for all n > 1.

Herschfeld's Convergence Theorem

An infinitely nested radical converges if and only if there is some such that for all.

Proof of "if"

We observe that the sequence in given by is monotonically increasing. Therefore if we exhibit an upper bound for the sequence then the sequence must converge by the Monotone convergence theorem. We do this:
where is the golden ratio.

Proof of "only if"

If the sequence given by is unbounded then the sequence is also unbounded. Therefore it must diverge.