A Brief Discussion on Nth Roots

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Aaron SeefeldtMath 646 April 18, 2006

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A Brief Discussion on nth Roots

The origins of inquiry into nth roots can be traced back to ancient Egyptians circa

3000 BC, as the pyramid architects would likely have used approximations of roots in

their geometric constructs. Although it is commonly believed that the Pythagorean

Theorem was known to the Egyptians at that time, nothing exists to substantiate this

explicitly. However, a papyrus scroll dating from 1650 BC (transcribed by Ahmes)

eluded to the Pythagorean Theorem by describing how to calculate the area of an

isosceles triangle. This and other anonymously extant information almost certainly

flowed into nearby cultures.

Notable and recorded improvements later occurred in the cradle of civilization.

The Babylonians, during the era 2000 600 BC, developed a method to calculate the

square root of a number that essentially used an iterative process of geometric means.

If ax = is the desired root, and if 1a is a first approximation, let1

1a

ab = . Thus the next

approximation would be )(2

1112 baa += and the next one would be )(

2

1223 baa += . This

process is equivalent to a two-term approximation of the binomial series. Because it is

likely that the Babylonians were interested in practical applications, they never extended

this algorithm to an infinite process. They later developed extensive tables of square

roots that survive to this day on clay tablets, and even developed what we would now

call logarithmic tables (in multiple numeric bases).

These Mesopotamians even had schemes to determine positiveroots of

quadratic and cubic equations, baxx =2 and cbxax =+ 23 respectively, but they had

no understanding of negative roots. It is believed that the mathematical knowledge that


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the Babylonians cultivated later spread to Hellenistic Greece and the Hindus in south-

east Asia.

The Pythagorean Theorem aside, the Pythagoreans are often credited with

discovering irrational numbers from the realization that the ratio between the side and

diagonal of a square cannot be a ratio of any two whole numbers. The consequence

being that 22 =x cannot have a rational root as an answer. However, the

Pythagoreans did not continue to develop this concept of irrational numbers. Would

they have known that pi is irrational? It is extremely unlikely. Although Archimedes

himself had calculated pi to two decimal places, it was not until the 18th

century that

Lambert finally proved pi to be irrational. In any case, Euclid also had an understanding

of roots of quadratic polynomials, for he developed a geometric method for solving

them.

Such discoveries were by no means limited to the West, even though much of

Europe went through a period of intellectual stagnation after the collapse of Rome. In

the 11th century, the Chinese mathematician Jia Xian developed a method for

determining the nth root of an integer, and then using that same process to find the nth

root of a polynomial of arbitrary degree. His method basically uses binomial coefficients

from Pascals triangle, but it is merely a special case of the general method derived by

Hornercenturies later.

About the same time, and after their aggressive expansion out of Arabia, Islamic

mathematicians also made important contributions. Although they had no firm

understanding of negative roots (or negative numbers really), they still laid substantial

groundwork for algebra. In the 9th century, al-Khwarizmi used techniques analogous to


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his ancient predecessors to find positive roots of quadratic equations. Later in the 12th

century, al-Khayyam made progress in solving cubic and higher degree polynomials,

and approximating irrational roots.

The mathematical dormancy of Europe was broken during the Renaissance,

probably because of the translation of Arabic texts into Latin. In the early 15th century,

the first, general solution to a cubic equation was developed by Cardano, followed by

the general solution to a quartic polynomial by Ferrai. Interestingly, however, the

general solution to a 5th degree polynomial and its roots remained much more elusive.

It was during this same era the two lingering issues continued to resurface again

and again. Negative numbers as roots were still regarded with skepticism, let alone the

strange roots of 222 + xx , which are 11 + and 11 . Descartes himself

referred to such negative roots as false, but it appears that he was aware that a

polynomial of degree n will have n roots, and developed a nomenclature to distinguish

between real and imaginary roots.

The groundwork for complex roots was initially laid by de Moivre when he noticed

the similarity between multiplying two complex numbers together, and the trigonometric

identity )sin()sin()cos()cos()cos( yxyxyx =+ . Later, Euler pieced much of the puzzle

together with his famous formula, but he himself lacked a deep understanding of what

complex numbers actually were, despite their usefulness.

Much of the work of the previous centuries eventually manifested itself in the

Fundamental Theorem of Algebra: every polynomial equation of degree n with complex

coefficients has n roots in the complex numbers. Although a few famous names

attempted proofs of the FTA, such as Leibnitz, DAlembert, Euler, and Lagrange, the


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first successful proof was done by Gauss in the 19 th century. The FTA and its

subsequent corollaries brought a great deal of closure to the angst of the preceding

centuries.

The determination of nth roots is easy to take for granted in modern time with the

advent of computers, calculators, and established theoretical background. Methods like

the recently invented shifting nth root algorithm(similar to long division) that is used to

determine the nth root of any positive real number make us forget the toil of ancient so

many ancient mathematicians.


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Bibliography

1. Greek Mathematical Thought and the Origin of Algebra. Jacob Klein. 1968,

Massachusetts Institute of Technology Press. Cambridge, MA. ISBN 0-486-27289-3.

2. The Norton History of the Mathematical Sciences. Ivor Grattan-Guinness. 1997,

W. W. Norton & Company, Inc. New York, NY. ISBN 0-393-04650-8.

3. A History of Mathematics. Carl Boyer. 1991, 2nd ed. John Wiley & Sons. New

York, NY. ISBN 0-471-54397-7.

4. Math Through the Ages: A Gentle History for Teachers and Others. Berlinghoff,

Gouvea. 2004. Oxton House Publishers. Farmington, ME. ISBN 0-88385-736-7.

5. Math & Mathematicians: The History of Math Discoveries Around the World. Leo

Bruno. 1999. UXL, Farmington Hills, MI. ISBN 0-7876-3814-5.

6. God Created the Integers: The Mathematical Breakthroughs that Changed History.

Stephen Hawking. 2005. Running Press Book Publishers. Philadelphia, PA. ISBN 0-

7624-1922-9.

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A Brief Discussion on Nth Roots