What Are Infinite Surds?
A surd is a sum with one or more irrational number expressed with a radical sign as addends. Examples are 1+√3, √2+√3, and √(1+√(1+√1)). Therefore, an infinite surd has an infinite number of such addends. An example is in the diagram.
How to Solve
The strategy for solving for the value of an infinite surd is to take advantage of repeating patterns. The sum is some unknown value, so set it to unknown x. Then see if x appears within the pattern of the surd, and replace it with an x, with the aim to replace an infinite number of terms with it. With x's on both sides of the equation, x can be solved for.
Surds of Unknown Numbers
The same approach can be used to solve surds that don't have numbers plugged in. In the diagram, k is an unknown or undetermined number, but the form of the equation is the same as above. (The quadratic formula is used here to solve the third line.) Note that the positive value of the radical is the only answer possible, since x is clearly positive. Note that if k=1, the result is the Golden Ratio.
Geometric Infinite Surd
The arguments of a surd need not be constant. They can increase exponentially. Amazingly, such an expression still converges instead of going to infinity. Note that the Golden Ratio again crops up in the answer.
In regard to the question of convergence, it may seem that, since 1 < √x for 1 < x, then these infinite surds should not be converging. After all, some power of a number greater than 1 is what is being added to the end as the surd is extended, right?
Not really. Because each additional radical is added to another number BEFORE the next radical to the left is applied, the addition of a new term is tempered tremendously, so that it is an addition of much less than 1.
For example, √25 = 5, but √26 ≈ 5.10 -- an increase of much less than 1.
The method of finding the value of infinite surds above can be extended to other infinite problems, for example infinite fractions and infinite decimals. Note that the Golden Ratio comes up yet again.
The Golden Ratio
At this point, one may wonder why the Golden Ratio keeps coming up, if it's peculiar to surds, or even what it is. It is defined as follows. Suppose one divides a line segment A into two pieces, B and C, where B is the larger of the two pieces. Suppose the break in A was positioned so that the ratio of B to C equals the ratio of A to B. That ratio is the Golden Rule. It is not peculiar to infinite surds. It also comes up in infinite fractions, as shown above. It comes up in geometry (especially the pentagon), number sequences, and even biology. Why it comes up so frequently is probably due to the simplicity of its definition. A number that solves equations as simple as x^2 - x - 1 = 0 and 1/x = x - 1 (which follow immediately from the line segment definition) is bound to crop up occasionally in number patterns and geometry.
- William Dunham; Journey Through Genius; 1991
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