Cubes and cube roots

Summary: Cubes and cube roots have been the subject of classical problems in mathematics, some of which were not solved for centuries.

Cubes and cube roots of numbers have played an important part in the development of mathematics. Middle school students are taught cubes and cube roots in order to solve equations and to calculate volumes of solids. In calculus, the cube root function is a common example of a function that is continuous everywhere but has an infinite derivative at one of its points. In addition, the cube function is an example of a function that is strictly increasing everywhere but has a point where the derivative is zero. The cube rule relates the percentage of popular vote in an election with the expected percentage of seats won in a two-party election. The power needed to overcome wind resistance is directly proportional to the cube of the wind speed. One model shows the heart rate in mammals is inversely proportional to the cube root of the weight of the animal. People in many different cultures have studied cubes and cube roots, and numerous interesting stories are found in its history. These simple objects have also generated many new ideas and new fields of mathematics.

Definition

To cube a particular number x, multiply it by itself 3 times—this is denoted x³. If x is a number such that x³=y for some other number y, then x is a cube root of y, written as x=∛y̅. Since (-5)³=-5×-×-5=-125, -5 is the cube root of -125, and the notation is

The cube of any real number is unique; however, every real number has exactly one cube root that is a real number and two cube roots that are complex numbers.

Early History

As with squares, the earliest uses of cubes of numbers involved common geometric objects, specifically the cube, which is a three-dimensional object with six sides, all of which are congruent squares. The volume of a cube is the cube of the length of one of its sides. The volume of a sphere is directly proportional to the cube of its radius. One of the classical problems in Greek mathematics was the problem called “Duplication of the Cube.” The problem was to find the length of an edge of a cube that has double the volume of a given cube using the tools of the time, the ruler and compass. It is now known that if x is the length of the side of the given cube, then ∛2×x is the length of the cube with twice the volume. One possible origin of this problem is that, in 430 b.c.e., it was proclaimed through the oracle at Delos that the cubical altar to Apollo was to be doubled in volume in order to alleviate a plague that had befallen the people. Another possibility is that the Pythagoreans successfully doubled the square and doubling the cube was a natural extension. In any event, many great mathematicians throughout history worked on this problem, and, in the nineteenth century, it was proven that a solution was impossible.

Cube roots can be exact numbers if the cube root is an integer or a fraction. However, the cube root of most numbers is irrational (it has a infinite non-repeating decimal expansion) and its value can only be approximated. The easiest method to approximate the real cube root of a real number is to raise the number to the 1/3 power on a calculator. Obviously, the calculator is a recent invention, and other methods have been developed for approximating a cube root. Some of the earliest known methods are found in the Chinese text Nine Chapters on the Mathematical Art (c. first century c.e.) and in the book Aryabhatiya by the Indian mathematician Aryabhata (b. 476 c.e.). Both methods use the formula (a+b)³=a³+3a²b+3ab²+b³ repeatedly to generate the successive digits of the cube root. Approximations to cube roots can also be computed with the Chinese abacus, the suànpán, which dates to 200 b.c.e. In many cases, scribes would create tables of cube roots, which people would use to look up values for use. Barlow’s Tables, named for mathematician Peter Barlow, who originally published in 1814, give the value of cubes and cube roots to nine decimal places and are still in print in the twenty-first century. Recreational mathematicians have found it fun to devise ways to compute cubes and approximations to cube roots in their head without outside assistance.

Cubic Equations

Cubic equations are equations that involve positive integer powers of x where the highest power is 3. Mathematicians have been trying to solve these equations from the earliest times. The Babylonian text BM 85200 (c. 2000 b.c.e.) contains many problems that compute the volume of an excavated rectangular cellar by setting up and solving a cubic equation. Another Babylonian tablet contains, among other things, a table of integers and the sum of each integer’s square and cube, and it was presumably used to solve cubic equations.

Archimedes of Syracuse (c. third century b.c.e.) considered the problem of passing a plane through a sphere such that the volumes of the two pieces had a certain ratio. This problem gives rise to what would be a cubic equation. A manuscript, thought to have been written by Archimedes, was found centuries later that gave a detailed solution to the problem that involved finding the intersection of a parabola with a hyperbola. Omar Khayyam (c. eleventh century c.e.) was the first to find a positive root of every cubic equation having one. Before this time, numbers were thought of as specific quantities of objects, so very little was done with negative numbers—and certainly not complex numbers. As with Archimedes, Khayyam’s solutions involved intersecting conic sections.

By the fifteenth and sixteenth centuries, negative numbers and zero were accepted, and many of the Greek mathematical texts were translated to Latin. The field of algebra had been developed, and people could study equations as expressions with variables that can be manipulated (as is done in the twenty-first century). As the solution of the general quadratic equation had been discovered, Italian mathematicians focused their attention to the solution of the general cubic equation ax³+bx²+cx+d=0. During this period, academic reputations and employment were based on public problem-solving challenges, and discoveries were kept secret so they could be used to win one of these challenges.

The solution of certain cubic equations provided the backdrop to one of the more entertaining chapters in the history of mathematics. On his deathbed in 1526, Italian mathematician Scipione del Ferro told one of his students, Antonio Maria Fior, how to solve a specific type of cubic equation. Nine years later, Fior submitted 30 cubic equations of this type to mathematician Niccolo Tartaglia in a public challenge. During the contest, Tartaglia himself discovered the solution and won the contest. After hearing of the contest, Girolamo Cardano contacted Tartaglia to inquire about his method. Tartaglia told him his solution, only after Cardano agreed to keep it secret as Tartaglia indicated he was going to publish it (thinking he was the first to discover it). Years later, Cardano found out that del Ferro actually discovered the formula and published it as del Ferro’s method in addition to solutions to the cubic equation in all cases that he and his assistant, Lodovico Ferrari, discovered. Tartaglia was extremely angry and felt Cardano had broken his promise. In the twenty-first century, the formula for the solution of the cubic is known as the Cardan(o)–Tartaglia formula.

Uses and Applications

The cubic equation also played an essential role in the formulation of complex numbers. In his 1572 text, Algebra, Rafael Bombelli considered the equation x³=15x+4. Applying the formula of Tartaglia and Cardano, one obtains a solution

However, Bombelli knew the solution was actually 4 and that, somehow, the square root of -121 could be manipulated in a way to reduce this expression to 4. He developed an algebra for working with these roots of negative numbers (thought to be of no use to earlier mathematicians), and complex numbers and the field of complex analysis was born. With complex numbers, one can show that any cubic equation has exactly three solutions, two of which must be complex and one real.

In 1670, it was discovered that French mathematician Pierre De Fermat claimed that for all natural numbers n>2 there are no nontrivial solutions of positive integers a, b, c such that aⁿ+bⁿ=cⁿ. Andrew Wiles proved this theorem in 1994 using objects called “elliptic curves.” Elliptic curves are defined by a cubic equation of the form y²=x³+ax+b, whose graph has no cusps or self-intersections. These curves are studied in the twenty-first century and used in both number theory and cryptography (the study of coding information). Even though Fermat’s equation has no positive integer solutions for n=3, other problems involving sums of cubes have been studied.

In 1770, Edward Waring proposed the following question: for every positive natural number k, does there exist a natural number s such that every natural number n can be written as the sum of at most s numbers which are kth powers? If k=3, the question becomes: can every positive number be written as a sum of at most s cubes? Some examples are 5=1³+1³+1³+1³+1³ and 23=2³+2³+1³+1³+1³ +1³ +1³+1³+1³.

As 23 shows, one requires at least 9 cubes. In 1909, David Hilbert proved that 9 is the maximum number of cubes that are required for any positive natural number. The Waring-Goldbach problem asks a similar question, except it requires at most s cubes of prime numbers. Some progress has been made, but this question remains unsolved as of 2010.

One of the more interesting recent mathematicians is Srinivasa Ramanujan from India (1887–1920). He was mostly self-educated and was able to prove theorems in number theory that shocked one of the eminent mathematicians of the time, G. H. Hardy. Once when Hardy visited Ramanujan, he mentioned he arrived in a cab numbered 1729, which did not seem very interesting. Ramanujan responded that 1729 is a very interesting number in that it is the smallest positive integer that can be represented by a sum of two cubes in two different ways, 1729=1³+12³=9³+10³, which is correct. The taxicab numbers are generalizations of this idea. The nth taxicab number, denoted Ta(n), is the smallest positive integer that can be written as two different cubes in n different ways. By Ramanujan’s comment, Ta(2)=1729. It is also true that Ta(1)=2, since 2=1³+1³ and Ta(3)=87,539,319, since

The first 6 taxicab numbers are known, but Ta(7) and beyond are all unknown as of 2010.

Bibliography

Burton D. A History of Mathematics: An Introduction. 7th ed. New York: McGraw-Hill, 2011.

Dunham, William. Journey Through Genius, New York: Penguin Books, 1991.

Katz, Victor. “The Roots of Complex Numbers.” Math Horizons 3 (1995).

Washington, Lawrence. Elliptic Curves: Number Theory and Cryptography. 2nd ed. Boca Raton, FL: CRC Press, 2008.