Infinite Surds

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  • Topic: Real number, Nth root, Indian mathematics
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Infinite Surds
Around 800 b.c.e, a young Indian scholar by the name of Baudhayana sat in his household studying manuscripts of previous mathematicians. He soon began his own research in mathematics and stumbled across a concept used in common mathematics today. This concept was added to a series of texts known as the Shulba Sastras. Today, the concept added to these books is known as square roots. Much like subtracting a number is the opposite of adding a number, the square root of a number is the opposite of squaring a number. For example, 32=9 ([pic]), likewise [pic]=3.

A radical, another name for a square root, comes in different forms. For example, if you change the root of a radical to a 3 rather than a 2, it becomes a cube root. Instead of squaring a number, 32=9, you will cube the number, 33=27 or in other words, 3*3*3=27. Therefore, you will have to cube root it, [pic]. The square roots of most numbers don’t always come out even. The number 9, is an example of a perfect square, because the square root of 9 is 3, a rational number. On the other hand, the square root of 8, does not come out even, when calculated through a calculator, the answer is 2.828427125, an irrational number. The exact values of these irrational numbers cannot be expressed in decimal form, and must be left in radical form because the decimal values are rounded and not completely accurate. These irrational radical numbers are called surds.

Surds can also be represented in series. In mathematics, given an infinite sequence of numbers (an ), a series is informally the result of adding all those terms together: a1 + a2 + a3 + · · ·. Therefore an example of an infinite series would be, [pic]as you can see the series increases by 1/2n each time. This can also be applied to surds as well, forming infinite surds. The expression [pic]would represent a basic infinite surd. These series are much more complex than basic series. Although this may be the case, the two are not that different after all. The infinite surd shown above is nothing but, a simple series of radicals when looked at piece by piece. [pic]

A general formula can be made for this infinite surd. When looking at the value for a2, a third of the series is the same as a2. Specifically, [pic], can also be represented as [pic] because [pic], same applies with a3, and a4. Therefore this statement can be made in terms of an rather than a2. The formula would technically be, [pic].

Like all series, an infinite surd has a pattern. When looking at the first 10 terms of the infinite surd [pic], your table of values turns out to be: [pic]

With our table of values, as well as our equation, ([pic]) we can create a graph showing the relationship between n and a2.

We can observe that as n increases, [pic] also increases but each time less than before, suggesting that at a large certain point of n it stops increasing and just follows in a straight line. To find the exact infinite value for this sequence we would use the equation and rearrange in the correct way to find the value.

The exact value for the surd is 1.618033989. On the other hand, the second value did not fit into the problem because it was negative.

This is a basic pattern seen throughout infinite surds. No matter what number or term is used in the surd, the process and the patterns will be similar. For instance, in the infinite surd, [pic] the pattern within the table of values is similar to that of the first infinite surd constructed.


In the previous data table, the values for an began to level at approximately 1.6, where as this time around, the values began evening out, around 2. Also similar to the 1st infinite surd, the equation to this has a similar format. The true difference when calculating the formula between the first infinite surd and this one, is the replacement of the number 2, with the number 1. Though when looking at the...
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