Throughout the years, the history of mathematics has taken its fair share of changes. It has stretched across the world from the Far East, migrating into the Western Hemisphere. One of the most fundamental and key principles of mathematics has been the quadratic formula. Having been used in several different cultures, the formula has been part of the base of mathematics theory. The general equation has been derived from many different sources, most commonly: ax2 + bx + c = 0, with x being the variable and a, b, and c its respective constant terms.
Though this is how modern mathematics perceives the equation, different symbols and notations have been used to represent the formula. Beginning in the Before Christ era, the Babylonians were the first to have been recorded demonstrating the equation, circa 400 BC. The form most mathematics students use today is: To solve a quadratic equation the Babylonians essentially used the standard formula, with the a term being included in the x2 variable. They considered two types of quadratic equations, namely: x2 + bx = c and x2 – bx = c Here b and c were positive but not necessarily integers.
The form that their solutions took was, respectively: x = [(b/2)2 + c] – (b/2) and x = [(b/2)2 + c] + (b/2). Notice that in each case this is the positive root from the two roots of the quadratic and the one that will make sense in solving “real” problems. For example problems which led the Babylonians to equations of this type often concerned the area of a rectangle. For example if the area is given and the amount by which the length exceeds the width is given, then the width satisfies a quadratic equation and then they would apply the first version of the formula above (website one).
The efforts the Babylonians made at using this method were far from futile and, actually, served a very important purpose. It was an important task for the rulers of Mesopotamia to dig canals and to maintain them, because canals were not only necessary for irrigation but also useful for the transport of goods and armies. The rulers or high government officials must have ordered Babylonian mathematicians to calculate the number of workers and days necessary for the building of a canal, and to calculate the total expenses of wages of the workers.
There are several Old Babylonian mathematical texts in which various quantities concerning the digging of a canal are asked for. They are YBC 4666, 7164, and VAT 7528, all of which are written in Sumerian … , and YBC 9874 and BM 85196, No. 15, which are written in Akkadian … . From the mathematical point of view these problems are comparatively simple (Muroi). The Babylonians used their mathematics not in the way we do today, by teaching, but by building their civilization into what it became.
Muroi speaks of how Babylonian mathematics helped create a society and how it helped the Mesopotamian region become as fertile as it was. It is for this reason that they only used positive forms in their answers. Had their use for mathematics been for reasons other than land, monetary, and other non-scientific reasons, they would have had to conclude their method with a negative result. Moving along the historical timeline, the Greeks were the next prominent mathematics society. Euclid of Alexandria is the most prominent mathematician of antiquity best known for his treatise on mathematics The Elements.
The long lasting nature of The Elements must make Euclid the leading mathematics teacher of all time. However little is known of Euclid’s life except that he taught at Alexandria in Egypt. Proclus, the last major Greek philosopher, who lived around 450 AD wrote: Not much younger than these [pupils of Plato] is Euclid, who put together the “Elements”, arranging in order many of Eudoxus’ theorems, perfecting many of Theaetetus’s, and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors.
This man lived in the time of the first Ptolemy; for Archimedes, who followed closely upon the first Ptolemy makes mention of Euclid, and further they say that Ptolemy once asked him if there were a shorted way to study geometry than the Elements, to which he replied that there was no royal road to geometry. He is therefore younger than Plato’s circle, but older than Eratosthenes and Archimedes; for these were contemporaries, as Eratosthenes somewhere says. In his aim he was a Platonist, being in sympathy with this philosophy, whence he made the end of the whole “Elements” the construction of the so-called Platonic figures (website two).
Though Euclid is most famous for his writings on geometry, it is the quadratic formula that has been at the forefront of problems dealing with area, and in turn, dimensional objects. 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 the root of a quadratic equation. Euclid had no notion of equations or coefficients but worked with purely geometrical quantities. The quantities he used were only pieces of the puzzle.
The great mathematician developed his theories himself, using the principles of the quadratic equation without actually using a derivation of the formula. Continuing the Greek methods were the Hindu mathematicians. Most notably, Brahmagupta, who lived from 598 665 BC, developed the first system of numbers that used zero. This system, a treatise called Brhma-sphuta-siddhnta, stated that a-a=0, and also considers the fractions x/0, which he sets equal to 0 for x=0 and otherwise calls nought, a term of uncertain meaning.
Not only is he the first mathematician to use zero, but he is also the first to determine a result undefined. Unlike most European algebraists of the Middle Ages, he recognized negative and irrational numbers as possible roots of an equation. Brahmagupta distinguished twenty arithmetical operations (logistics), including the extraction of roots and the solution of proportions, and eight measurements (determinations). In this fine classification of mathematical procedures, he also listed four methods for multiplication, and five rules for reducing a rational expression to a single fraction (website three).
His mathematics was a mixture of concrete problems and abstract formulas (determination of the partial sums of the series of perfect squares and of perfect cubes), which he sometimes gave in form of verses, or by means of numerical examples. Brahmagupta’s writings were taken to Baghdad, from where they influenced the development of the exact sciences in the Arab world. Traces of his results can be found in al-Khwarizmi (van der Waerden, 1985, p. 14). Since the Arabs had not learned completely of the Hindu methods, they still did not have any reasoning involving zeroes or undefined terms.
The most notable of the Arabic mathematicians was Abu Ja’far Muhammad ibn Musa Al-Khwarizmi (usually referred to as al-Khwarizmi). Al-Khwarizmi was born around AD 790 near Baghdad, and died around 850 BC. His most important contribution, written in 830 BC, was Hisab al-jabr wal-muqabala. From the al-jabr in the title we get algebra. The treatise develops a system for the solutions of quadratic expressions including geometric principles for completing the square. Al-Khwarizmi also gave a classification system for quadratics.
He devotes a chapter to each chapter in his treatise and gives methods in solving each differently: Six Types of Quadratics 1. Squares equal to roots (x = square root of 2) 2. Squares equal to numbers (x = 2) 3. Roots equal to numbers (square root of x = 2) 4. Squares and roots equal to numbers (x + 3x = 25) 5. Squares and numbers equal to roots (x + 1 = 9) 6. Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example that is a geometrical completing the square (website four).
The geometric proof by completing the square follows. Al-Khwarizmi starts with a square of side x, which therefore represents x2 (Figure 1). To the square we must add 10x and this is done by adding four rectangles each of breadth 10/4 and length x to the square (Figure 2). Figure 2 has area x2 + 10 x which is equal to 39. We now complete the square by adding the four little squares each of area 5/2 5/2 = 25/4. Hence the outside square in Fig 3 has area 4 25/4 + 39 = 25 + 39 = 64. The side of the square is therefore 8.
But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3. (website four) Roots and numbers equal to squares (3x + 4 = x) The term Renaissance, adopted from the French equivalent of the Italian word rinascita, meaning literally “rebirth,” describes the radical and comprehensive changes that took place in European culture during the 15th and 16th centuries, bringing about the demise of the Middle Ages and embodying for the first time the values of the modern world. The consciousness of cultural rebirth was itself a characteristic of the Renaissance.
Italian scholars and critics of this period proclaimed that their age had progressed beyond the barbarism of the past and had found its inspiration, and its closest parallel, in the civilizations of ancient Greece and Rome. A new phase of mathematics began in Italy around 1500. In 1494 the first edition of Summa de arithmetica, geometrica, proportioni et proportionalita, now known as the Suma, appeared. It was written by Luca Pacioli although it is quite hard to find the author’s name on the book, Fra Luca appearing in small print but not on the title page.
In many ways the book is more a summary of knowledge at the time and makes no major advances. The work gives a summary of the mathematics known at that time although it shows little in the way of original ideas. The work studies arithmetic, algebra, geometry and trigonometry and, despite the lack of originality, was to provide a basis for the major progress in mathematics that took place in Europe shortly after this time. As stated, the Summa was: … not addressed to a particular section of the community.
An encyclopaedic work (600 pages of close print, in folio) written in Italian, it contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of Euclid’s geometry. He admitted to having borrowed freely from Euclid, Boethius, Sacrobosco, Fibonacci, … The geometrical part of Pacioli’s Summa is discussed in detail by Glushkova and Glushkov. The authors write:
The geometrical part of L Pacioli’s Summa [Venice, 1494] in Italian is one of the earliest printed mathematical books. Pacioli broadly used Euclid’s Elements, retelling some parts of it. He referred also to Leonardo of Pisa (Fibonacci). Other interesting aspects of the Summa was the fact that it studied games of chance. Pacioli studied the problem of points although the solution he gave is incorrect. In general, Pacioli never actually devised his own method of solving the quadratic equation, but instead published a works dealing with the history of certain aspects of mathematics, including all the great mathematicians ideas.
Overall, the quadratic formula has actually shaped civilization into the way it is today. The Babylonians used it to irrigate their land, divide out funds to pay workers, and even mobilize armies. The formula to them was not a way of learning or teaching, but a way of life. Euclid was a man of geometry. He theorized about area, volume, shapes, and even dimensions, finding a way to calculate all of the above. The quadratic formula only guided him in his writings and research, though he did not even know he was using a formula, per say.
The Hindu and Arabic approaches were, for the most part, directly related to teaching and understanding. They used different methods, including completing the square, to get their results. All of these works and ideas progressed into the renaissance. From then on the idea of the quadratic formula were only summarized and taught so that the rest of the world would know how mathematics works. The equation is one of the most prominent ideas in mathematics, and is the center of foundation of mathematics itself: algebra.
High school and college students may be taking courses of higher mathematics but have all, at one point in their studies, learned and used the quadratic formula. The idea that mathematics has changed over time is opinionative but very provable. From the beginning of time the human race has needed ways to count and manipulate quantities. Great minds throughout the years have studied their predecessors and developed theories molding mathematics and the quadratic formula into what it is today.