Wednesday, September 4, 2013

Mathematics -- Part Three: Algebra

Algebra can be considered as doing computations similar to that of arithmetic with non-numerical mathematical objects, often represented by letters or other abbreviations. Initially, these objects represented either numbers that were not yet known (unknowns) or unspecified numbers (indeterminate or parameter), allowing one to state and prove properties that are true no matter which numbers are involved. For example, in the quadratic equation,

ax2 + bx + c = 0

a, b, and c are parameters called “coefficients” and x is the unknown. Solving this equation amounts to computing with the variables to express the unknowns in terms of the coefficients. Then, substituting any numbers for the coefficients, gives the solution of a particular equation after a simple arithmetic computation.

You can consider algebra as “abstract” arithmetic where some of the numbers are replaced with letters. You then perform mathematical or arithmetic operations on the letters. This often requires you understand the rules of arithmetic very clearly to perform operations that would be correct for all possible values of the “numbers” including negative values, zero, and even Complex numbers. If given values for the parameters or coefficients, then solve for x. That is algebra.

Another view is that algebra can be used to create a general solution, often called a “formula.” Then all you need to do is substitute actual values for the letters and solve the formula (using arithmetic) to obtain a final result. For example, the solution of the quadratic equation is

This is the “general solution” to the quadratic. This equation and solution is well-known to all graduates of an introductory algebra class. It is part of a group of equations called “polynomials.” Since the highest power in the quadratic equation is 2, this is a second degree polynomial. Less than 2 and it isn’t a “poly.” So the quadratic is the most basic equation of a “non-linear” type. Non-linear means that, if you graph the equation, it is not a straight line. (We will get into graphing equations when we talk about analytical geometry and the connection between the math of numbers and the math of shapes. That’s later in this series.)

There are many ways to solve a quadratic from factoring to completing the square. (Square is where the name “quadratic” came from.) The formula or “general solution” is derived from the latter method. Most remember from that beginning algebra class that there are two solutions or “roots” to a quadratic equation. In the solution formula they come from the “plus-minus” with each sign providing a solution.

There is a proven theorem in mathematics that is so significant it is called the “Fundamental Theorem of Algebra.” It states that, every non-zero, single-variable, degree n polynomial has exactly n roots. (Actually it is a little more complicated than that since the theorem addresses Complex coefficients, but we won’t get into that much detail.)

What this fundamental theorem means is that all second degree polynomial equations (those with the highest power of 2) have 2 roots. In addition, a polynomial equation with cubes will have 3 roots. One with the seventh power will have 7, etc. (It says a little more about the field of complex numbers being algebraically closed, but we REALLY won’t get into that.)

Additionally, it is not fundamental for modern algebra; its name was given at a time when the study of algebra was mainly concerned with the solutions of polynomial equations with real or complex coefficients. And that is exactly where we want to go. A little history of math that bridges the time from the ancient Greeks to the modern rebirth of math during the enlightenment.

Even though the Babylonians didn't have any notion of what an “equation” is, they found the first algorithmic approaches to problems, which would give rise to a quadratic equation today. Their method is essentially one of completing the square. However all Babylonian problems had answers which were positive (more accurately unsigned) quantities since the usual answer was a length.

The first known solution of a quadratic equation is the one given in the Berlin papyrus from the Middle Kingdom (ca. 2160-1700 BC) in Egypt.

In about 300 BC Euclid developed a geometrical approach which, although later mathematicians used it to solve quadratic equations, amounted to finding a length which in our notation was a root of a quadratic equation. Euclid had no notion of equation, coefficients, etc., but worked with purely geometrical quantities.

In his work, Arithmetica, the Greek mathematician Diophantus (ca. 210-290 AD) solved the quadratic equation, but giving only one root, even when both roots were positive. Hindu mathematicians took the Babylonian methods further. Aryabhata around 500 AD gave a rule for the sum of a geometric series that shows knowledge of the quadratic equations with both solutions.

So the general solution to the quadratic has been known since ancient times, before the concept of an equation had evolved and using somewhat sloppy and unproven methods, at least when compared to modern axiomatic-based math. But the method did work and was well known. It was often used in problems involving rectangles like surveying or planting of crops. It was a beginning.

This knowledge was preserved by the Arabs in the Middle East as civilization in Europe collapsed after the fall of Rome. The Hindus also continued to advance algebra, developing nearly modern methods that found both roots, even negative ones; but the Arabs were responsible primarily for preserving the Greek methods and introducing them to Europe after the end of the Dark Ages.

Even the word “algebra” comes from a bit of a mistranslation of Middle Eastern books. The word algebra is a Latin variant of the Arabic word al-jabr. This came from the title of a book, Hidab al-jabr wal-muqubala ("The Compendious Book on Calculation by Completion and Balancing"), written in Baghdad about 825 AD by the Arab mathematician Mohammed ibn-Musa al-Khowarizmi.

Abraham bar Hiyya Ha-Nasi, often known by the Latin name Savasorda, is famed for his book, Liber embadorum, published in 1145, which is the first book published in Europe to give the complete solution of the quadratic equation.

Next came the general solutions for a cubic equation, a polynomial of power 3. Scipione del Ferro (1465-1526), the Chair of Arithmetic and Geometry at the University of Bologna, is credited with solving cubic equations algebraically, but the story is somewhat more complicated. We believe that del Ferro could only solve cubic equation of the form

x3 + mx = n — no second power term.

However, without the Hindu's knowledge of negative numbers, del Ferro would not have been able to use his solution of the one case to solve all cubic equations. Remarkably, del Ferro solved this cubic equation around 1515 but kept his work a complete secret until just before his death, in 1526, when he revealed his method to his student Antonio Fior.

Fior was a mediocre mathematician and far less good at keeping secrets than del Ferro. Soon rumors started to circulate in Bologna that the cubic equation had been solved. Nicolo of Brescia, known as Tartaglia meaning "the stammerer," prompted by the rumors, managed to solve equations of the form x3 + mx2 = n, a slightly different case, and made no secret of his discovery.

Fior challenged Tartaglia to a public contest: the rules being that each gave the other 30 problems with 40 or 50 days in which to solve them, the winner being the one to solve most but a small prize was also offered for each problem. Tartaglia solved all Fior's problems in the space of 2 hours, for all the problems Fior had set were of the form x3 + mx = n, as he believed Tartaglia would be unable to solve this type. However only 8 days before the problems were to be collected, Tartaglia had found the general method for all types of cubics.

News of Tartaglia's victory reached Girolamo Cardan in Milan where he was preparing to publish Practica Arithmeticae (1539). Cardan invited Tartaglia to visit him and, after much persuasion, made him divulge the secret of his solution of the cubic equation. This Tartaglia did, having made Cardan promise to keep it secret until Tartaglia had published it himself. Cardan did not keep his promise. In 1545 he published Ars Magna the first Latin treatise on algebra.

Cardan noticed something strange when he applied his formula to certain cubics. When solving

x3 - 15x = 4

he obtained an expression involving √-121. Cardan knew that you could not take the square root of a negative number yet he also knew that x = 4 was a solution to the equation. He wrote to Tartaglia in August 1539 in an attempt to clear up the difficulty. Tartaglia certainly did not understand. In Ars Magna Cardan gives a calculation with "Complex" numbers" to solve a similar problem, but he really did not understand his own calculation which he said is "as subtle as it is useless."

During the 1500s, solving these complex equations became a form of entertainment and mathematicians would travel the country and hold side shows where they challenged the audience to give them a problem and they would solve it quickly. They were using these general solutions and the search was underway for easy ways to solve harder and harder problems. Who knows how many discoveries were made but kept secret to maintain the "magic" of the show.

In 1540, Cardan was given the following problem: Divide 10 into 3 parts: The parts are in continued proportion and the product of the first 2 is 6

This problem led to a quartic polynomial (power of 4) which Cardan was not able to solve. He gave it to his student, Lodovico Ferrari. Ferrari was the first to develop an algebraic technique for solving the general quartic. He applied his technique (which was published by Cardano ) to the equation

x4 + 6x2 - 60x + 36 = 0

Eventually a more general solution was refined. The general quartic equation is a fourth-order polynomial equation of the form

x4 + a3x3 + a2x2 + a1x + a0 = 0

Although this might not appear completely general as there is no coefficient for the first term, it actually is. You may also wonder where are the minus signs since all these polynomials look like they just use pluses. But the coefficients can be negative, which provides the minus signs.

The general solution for the “biquadratic” or “quartic” polynomial was much more complicated and reduced down to a combination of a linear equation (power of 1) and a quadratic. As this general solution was learned, the race was on to solve even higher power polynomials using general solutions.

Unlike quadratic, cubic, and quartic polynomials, the general quintic cannot be solved algebraically in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions, as rigorously demonstrated by Neils Henrick Abel (Abel's impossibility theorem) and by √Čvariste Galois in the early 1800s.

The theoretical work in devising a general, algebraic solution to these polynomials also provided proof that the mathematicians were at the end of the road. There are no general solutions to any higher power polynomials. The road shows had ended and the work shifted to the university and publications. However, further solutions to the quintic form were added in the late nineteenth century and even in 1994 new and more solutions to certain cases of quintic polynomials were added by Spearman and Williams. So these ancient puzzles are still being solved. More significant, however, was the advancements in algebraic theory that were driven by the old side show competitions and the work to put algebra on solid theoretical grounds.

There are lots of areas of algebra that I’ve skimmed over. There are problems with more than one unknown, systems of equations and problems with complex coefficients and other branches of math such as trigonometry.

As it developed, algebra was extended to other non-numerical objects, like vectors and matrices. Then, the structural properties of these non-numerical objects were abstracted to define algebraic structures like groups, rings, fields and algebras. Eventually, algebra was even extended to Geometry.

And that is our next destination: Geometry.

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