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Nested radical

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In algebra, a nested radical is a radical expression (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, and more complicated ones such as


Some nested radicals can be rewritten in a form that is not nested. For example,


Another simple 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[edit]

In the case of two nested square roots, the following theorem completely solves the problem of denesting.[2]

If a and c are rational numbers and c is not the square of a rational number, there are two rational numbers x and y such that

if and only if is the square of a rational number d.

If the nested radical is real, x and y are the two numbers

and where is a rational number.

In particular, if a and c are integers, then 2x and 2y are integers.

This result includes denestings of the form

as z 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 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 x = c and In the second case, and another coefficient must be zero. If one may rename xy as x 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 x and y, solving the first equation in x and y 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 (otherwise the right-hand side of the equation would be rational; but the left-hand side is irrational). As x and y 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 1 and implies thus that the considered equation is equivalent with
It follows by Vieta's formulas that x and y must be roots of the quadratic equation
its (≠ 0, otherwise c would be the square of a), hence x and y must be
and Thus x and y 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 c > 0. The equation shows that |a| > c. Thus, if the nested radical is real, and if denesting is possible, then a > 0. Then the solution is

Some identities of Ramanujan[edit]

Srinivasa Ramanujan demonstrated a number of curious identities involving nested radicals. Among them are the following:[3]



Landau's algorithm[edit]

In 1989 Susan Landau introduced the first algorithm for deciding which nested radicals can be denested.[5] Earlier algorithms worked in some cases but not others. Landau's algorithm involves complex roots of unity and runs in exponential time with respect to the depth of the nested radical.[6]

In trigonometry[edit]

In trigonometry, the sines and cosines of many angles can be expressed in terms of nested radicals. For example,


The last equality results directly from the results of § Two nested square roots.

In the solution of the cubic equation[edit]

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 (even though not obviously so) to one of the solutions 1, 2, or –3.

Infinitely nested radicals[edit]

Square roots[edit]

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 (the second solution x = −1 doesn't apply, under the convention that the positive square root is meant). This approach can also be used to show that generally, if n > 0, then

and is the positive root of the equation x2xn = 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 > 0,[citation needed]

which is the positive root of the equation x2 + xn = 0.

Nested square roots of 2[edit]

The nested square roots of 2 are a special case of the wide class of infinitely nested radicals. There are many known results that bind them to sines and cosines. For example, it has been shown that nested square roots of 2 as[7]

where with in [−2,2] and for , are such that for

This result allows to deduce for any the value of the following infinitely nested radicals consisting of k nested roots as

If , then[8]

These results can be used to obtain some nested square roots representations of . Let us consider the term defined above. Then[7]

where .

Ramanujan's infinite radicals[edit]

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(x) and squaring both sides gives us

which can be simplified to

It can then be shown that, assuming is analytic,

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 π[edit]

Viète's formula for π, the ratio of a circle's circumference to its diameter, is

Cube roots[edit]

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 x3xn = 0 for all n > 0. For n = 1, this root is the plastic ratio ρ, approximately equal to 1.3247.

The same procedure also works to get

as the real root of the equation x3 + xn = 0 for all n > 1.

Herschfeld's convergence theorem[edit]

An infinitely nested radical (where all are nonnegative) converges if and only if there is some such that for all ,[9] or in other words

Proof of "if"[edit]

We observe that

Moreover, the sequence is monotonically increasing. Therefore it converges, by the monotone convergence theorem.

Proof of "only if"[edit]

If the sequence converges, then it is bounded.

However, , hence is also bounded.

See also[edit]


  1. ^ Scheinerman, Edward R. (2000), "When close enough is close enough", American Mathematical Monthly, 107 (6): 489–499, doi:10.2307/2589344, JSTOR 2589344, MR 1766736
  2. ^ Euler, Leonhard (2012). Elements of algebra. Springer Science & Business Media. Chapter VIII.
  3. ^ Landau, Susan (July 16, 1993). "A note on 'Zippel Denesting'". CiteSeerX Retrieved August 23, 2023.
  4. ^ Berndt, Bruce; Chan, Heng; Zhang, Liang-Cheng (1998). "Radicals and units in Ramanujan's work" (PDF). Acta Arithmetica. 87 (2): 145–158. doi:10.4064/aa-87-2-145-158.
  5. ^ Landau, Susan (1992). "Simplification of Nested Radicals". 30th Annual Symposium on Foundations of Computer Science. Vol. 21. SIAM. pp. 85–110. CiteSeerX doi:10.1109/SFCS.1989.63496. ISBN 978-0-8186-1982-3. S2CID 29982884.
  6. ^ Gkioulekas, Eleftherios (2017-08-18). "On the denesting of nested square roots". International Journal of Mathematical Education in Science and Technology. 48 (6): 942–953. Bibcode:2017IJMES..48..942G. doi:10.1080/0020739X.2017.1290831. ISSN 0020-739X. S2CID 9737528.
  7. ^ a b Servi, L. D. (April 2003). "Nested Square Roots of 2". The American Mathematical Monthly. 110 (4): 326–330. doi:10.1080/00029890.2003.11919968. ISSN 0002-9890. S2CID 38100940.
  8. ^ Nyblom, M. A. (November 2005). "More Nested Square Roots of 2". The American Mathematical Monthly. 112 (9): 822–825. doi:10.1080/00029890.2005.11920256. ISSN 0002-9890. S2CID 11206345.
  9. ^ Herschfeld, Aaron (1935). "On Infinite Radicals". The American Mathematical Monthly. 42 (7): 419–429. doi:10.2307/2301294. ISSN 0002-9890. JSTOR 2301294.

Further reading[edit]