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,
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 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,
- |x| ≥ |y|,
(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
its (≠0, otherwise c would be the square of a), hence x and y must be
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| > √. Thus, if the nested radical is real, and if denesting is possible, then a > 0. Then, the solution writes
Some identities of Ramanujan
Other odd-looking radicals inspired by Ramanujan include:
This section needs expansion. You can help by adding to it. (February 2015)
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. Landau's algorithm involves complex roots of unity and runs in exponential time with respect to the depth of the nested radical.
The last equality results directly from the results of § Two nested square roots.
In the solution of the cubic equation
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
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 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(x) 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 π
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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
Proof of "if"
We observe that
Moreover, the sequence is monotonically increasing. Therefore it converges, by the monotone convergence theorem.
Proof of "only if"
If the sequence converges, then it is bounded.
However, , hence is also bounded.
- Euler, Leonhard (2012). Elements of algebra. Springer Science & Business Media. Chapter VIII.
- Landau, Susan (1993). "A note on 'Zippel Denesting'". CiteSeerX 10.1.1.35.5512. Cite journal requires
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- 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.
- Landau, Susan (1992). "Simplification of Nested Radicals". 30th Annual Symposium on Foundations of Computer Science. Journal of Computation. 21. SIAM. pp. 85–110. CiteSeerX 10.1.1.34.2003. doi:10.1109/SFCS.1989.63496. ISBN 978-0-8186-1982-3. S2CID 29982884. CS1 maint: discouraged parameter (link)
- 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. doi:10.1080/0020739X.2017.1290831. ISSN 0020-739X.
- Herschfeld, Aaron (1935). "On Infinite Radicals". The American Mathematical Monthly. 42 (7): 419–429. doi:10.2307/2301294. ISSN 0002-9890. JSTOR 2301294.
- Landau, Susan (1994). "How to Tangle with a Nested Radical". Mathematical Intelligencer. 16 (2): 49–55. doi:10.1007/bf03024284. S2CID 119991567. CS1 maint: discouraged parameter (link)
- Decreasing the Nesting Depth of Expressions Involving Square Roots
- Simplifying Square Roots of Square Roots
- Weisstein, Eric W. "Square Root". MathWorld.
- Weisstein, Eric W. "Nested Radical". MathWorld.