which arises in discussing the regular pentagon, and more complicated ones such as
- 1 Denesting nested radicals
- 2 In trigonometry
- 3 In the solution of the cubic equation
- 4 Infinitely nested radicals
- 5 See also
- 6 References
Denesting nested radicals
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 process is generally considered a difficult problem, although a special class of nested radical can be denested by assuming it denests into a sum of two surds:
Squaring both sides of this equation yields:
This can be solved by finding two numbers such that their sum is equal to a and their product is b2c/4, or by equating coefficients of like terms—setting rational and irrational parts on both sides of the equation equal to each other. The solutions for e and d can be obtained by first equating the rational parts:
For the irrational parts note that
and squaring both sides yields
By plugging in a − d for e one obtains
Rearranging terms will give a quadratic equation which can be solved for d using the quadratic formula:
Since a = d+e, the solution e is the algebraic conjugate of d. If we set
In some cases, higher-power radicals may be needed to denest the nested radical.
Some identities of Ramanujan
Other odd-looking radicals inspired by Ramanujan include:
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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
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 pi
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 a 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 > 0.
The value obtained for the infinite nested radical by converting to a polynomial equation and solving is valid only if the sequence of values, obtained by successively nesting more and more radicals, converges. For example, the above-considered expression
if convergent, is the limiting value of the process
starting from the initial value We have
Convergence requires that the absolute value of this expression be less than 1 in the neighborhood of the value of x given earlier that satisfies the corresponding polynomial equation. It turns out that this condition is that n > 3/4, which holds if, for example, we require the positive number n to be an integer. Then it is sufficient for convergence that the initial value be in the basin of attraction of the indicated stationary value of x.
- Landau, Susan (1993). "A note on 'Zippel Denesting'". CiteSeerX: 10
.1 .1 .35 .5512.
- Landau, Susan. "RADICALS AND UNITS IN RAMANUJAN'S WORK" (PostScript).
- Landau, Susan (1992). "Simplification of Nested Radicals". Journal of Computation. SIAM. 21: 85–110. doi:10.1109/SFCS.1989.63496. CiteSeerX: 10
.1 .1 .34 .2003.
- Landau, Susan (1994). "How to Tangle with a Nested Radical". Mathematical Intelligencer. 16: 49–55. doi:10.1007/bf03024284.
- Decreasing the Nesting Depth of Expressions Involving Square Roots
- Simplifying Square Roots of Square Roots
- Weisstein, Eric W. "Square Root". MathWorld.