Nested radical: Difference between revisions
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In 1989 [[Susan Landau]] introduced the first [[algorithm]] for deciding which nested radicals can be denested.<ref>{{cite book | citeseerx = 10.1.1.34.2003 | title = Simplification of Nested Radicals | authorlink = Susan Landau | first = Susan | last = Landau | publisher = [[Society for Industrial and Applied Mathematics|SIAM]] | journal = Journal of Computation | volume = 21 | year = 1992 | pages = 85–110 | doi = 10.1109/SFCS.1989.63496 | isbn = 978-0-8186-1982-3 }}</ref> Earlier algorithms worked in some cases but not others. |
In 1989 [[Susan Landau]] introduced the first [[algorithm]] for deciding which nested radicals can be denested.<ref>{{cite book | citeseerx = 10.1.1.34.2003 | title = Simplification of Nested Radicals | authorlink = Susan Landau | first = Susan | last = Landau | publisher = [[Society for Industrial and Applied Mathematics|SIAM]] | journal = Journal of Computation | volume = 21 | year = 1992 | pages = 85–110 | doi = 10.1109/SFCS.1989.63496 | isbn = 978-0-8186-1982-3 }}</ref> Earlier algorithms worked in some cases but not others. |
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Leonhard Euler (1707–1783) in his book ''Vollständige Algebra'', page 270. |
Leonhard Euler (1707–1783) in his book ''Vollständige Algebra'', page 270. |
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be: V(a+Vb) = Vx + Vy ... V stands for square root. |
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be: dicriminant (a*a-b)=d*d → Sol. exists iff d is a natural number. |
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His algorithm seems to work perfectly. |
His algorithm seems to work perfectly. |
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Revision as of 11:15, 13 May 2019
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
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 d and e 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:
which gives
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
then
However, this approach works for nested radicals of the form if and only if is a rational number, in which case the nested radical can be denested into a sum of surds.
In some cases, higher-power radicals may be needed to denest the nested radical.[citation needed]
Some identities of Ramanujan
Srinivasa Ramanujan demonstrated a number of curious identities involving denesting of radicals. Among them are the following:[1]
Other odd-looking radicals inspired by Ramanujan include:
Landau's algorithm
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.[3] Earlier algorithms worked in some cases but not others.
Leonhard Euler (1707–1783) in his book Vollständige Algebra, page 270.
be: V(a+Vb) = Vx + Vy ... V stands for square root.
be: dicriminant (a*a-b)=d*d → Sol. exists iff d is a natural number. His algorithm seems to work perfectly.
In trigonometry
In trigonometry, the sines and cosines of many angles can be expressed in terms of nested radicals. For example,
and
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 (even though not obviously so) 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 (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
Viète's formula for π, 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.
See also
References
- ^ Landau, Susan (1993). "A note on 'Zippel Denesting'". CiteSeerX 10.1.1.35.5512.
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(help) - ^ 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. Vol. 21. SIAM. pp. 85–110. CiteSeerX 10.1.1.34.2003. doi:10.1109/SFCS.1989.63496. ISBN 978-0-8186-1982-3.
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ignored (help)
Further reading
- Landau, Susan (1994). "How to Tangle with a Nested Radical". Mathematical Intelligencer. 16 (2): 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.
- Weisstein, Eric W. "Nested Radical". MathWorld.