Maggio: History of Algebra and its Implications for Teaching History of Algebra and its Implications for Teaching Jaime Maggio Fall 2020 MA 398 Senior Seminar Mentor: Dr.Loth Published by DigitalCommons@SHU, 2021 1 Academic Festival, Event 31 [2021] Abstract Algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities (letters and other general sym- bols) defined by the equations that they satisfy. Algebraic problems have survived in mathematical writings of the Egyptians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts. In this paper, we will discuss historically famous mathematicians from all over the world along with their key mathematical contributions. Mathe- matical proofs of ancient and modern discoveries will be presented. We will then consider the impacts of incorporating history into the teaching of mathematics courses as an educational technique. 1 https://digitalcommons.sacredheart.edu/acadfest/2021/all/31 2 Maggio: History of Algebra and its Implications for Teaching 1 Introduction In order to understand the way algebra is the way it is today, it is important to understand how it came about starting with its ancient origins. In a mod- ern sense, algebra can be described as a branch of mathematics concerned with finding the values of unknown quantities defined by the equations that they sat- isfy. Algebraic problems have survived in mathematical writings of the Egyp- tians and Babylonians. The ancient Greeks also contributed to the development of algebraic concepts, but these concepts had a heavier focus on geometry [1]. The combination of all of the discoveries of these great mathematicians shaped the way algebra is taught today. It can be important to consider how history is or can be incorporated into the teaching of it in school systems. In this paper, we will start off by discussing famous mathematicians from around the world and their contributions in section 2. In section 3 proofs of theorems discovered by some of these famous mathematicians, specifically from ancient times, will be presented. In section 4 more proofs will be presented, but these proofs will in- volve concepts from more modern times. Lastly, in section 5 we will discuss the implications for teaching. 2 Published by DigitalCommons@SHU, 2021 3 Academic Festival, Event 31 [2021] 2 History and Background 2.1 Greek Mathematicians One highly influential ancient Greek mathematician was Pythagoras. Pythago- ras lived during the sixth century B.C and was a great mathematician as well as a philosopher. Over time he gained many followers known as Pythagoreans. He has been given credit for and is most well known for his theorem known as the Pythagorean theorem [2]. This theorem states that the sum of the squares of the lengths of the two shorter sides of a right triangle is equal to the square of the length of the longer side (hypotenuse) of the triangle. In its most common form, the theorem states that a2 + b2 = c2 where a and b are the lengths of the legs of the right triangle and c is the length of the hypotenuse [3]. Not only is Pythagoras known for this work with right triangles, but the Pythagoreans are also known for the discovery of irrational numbers. Geometrically speaking, if AB and CD are two segments, they are considered commensurable if there is a unit EF such that both AB and CD are multiples of EF, so that AB : CD = m : n where m and n are positive integers. What the Pythagoreans discovered was that the side and the diagonal of a square are not commensurable. If the side of a square has length 1, this implies that the diagonal, which has length p p 2, is not commensurable with this side. In other words, 2 is not a rational number [2]. This proof will be shown in section 3 of the paper. Pythagoras cer- 3 https://digitalcommons.sacredheart.edu/acadfest/2021/all/31 4 Maggio: History of Algebra and its Implications for Teaching tainly expressed how the world is highly mathematical and initiated this new way of thinking [3]. He was able to make contributions that would aid in future algebraic discoveries. Another influential Greek Mathematician was Euclid. Euclid is one of the most famous mathematicians of all time. He lived in Alexandria Greece around 300 B.C. He was able to establish Geometry as a deductive science based on a small number of fundamental principles called axioms [2]. His major work where he presented these ideas is called Elements. Euclid's Elements is said to have been one of the most influential mathematical treatises. Although his work does not contain much original content, the order of propositions and overall logical struc- ture of his work is what made it stand out. Some of the most famous of the the- orems in the Elements are the following: The sum of the three angles of a triangle is equal to two right angles. The area of the square on the hypotenuse of a right angled triangle is equal to the sum of the areas of the squares on the other two sides. Euclid's Elements contained the "windmill proof" of the Pythagorean theorem. This proof used definitions regarding things such as lines, right angles, parallel lines and self evident postulates. He also included a proof of the converse of this theorem. 4 Published by DigitalCommons@SHU, 2021 5 Academic Festival, Event 31 [2021] 2.2 Persian Mathematicians Al-Khwarizmi was a key Persian mathematical figure who introduces the con- cepts of algebra into European mathematics. He was one of the most influential figures of his time and was a central figure in the separation of Algebra from Ge- ometry and its development as an independent discipline. He was able to clas- sify quadratic equations and even gave methods to find solutions for each type [2]. In 825 CE, Al-Khwarizmi discovered a general method that is similar to to- day's Quadratic Formula. He gave a method to solve any equation of the form aX2 + bX = c, where a,b and c are positive numbers [8]. He did not consider equations of the form aX2 + bX + C = 0 because he considered only positive numbers. If a,b, and c were all positive numbers then this equation would have no positive solutions. An Italian mathematician who we will talk about later on in the paper, Gerolamo Cardano, combined Al-Khwarizmi's solution with geom- etry to solve quadratic equations. Cardano allowed for negative solutions in his method. There was also a French mathematician, Rene Descartes, who actually published the Quadratic Formula as it exists today in his work. 2.3 Italian Mathematicians After discovering solutions to quadratic equations, mathematicians around the world were struggling to come up with a solution to third power equations. Sci- 5 https://digitalcommons.sacredheart.edu/acadfest/2021/all/31 6 Maggio: History of Algebra and its Implications for Teaching pione Del Ferro was an Italian mathematician who made great achievements in this area. He lived from 1465 to 1526 and was a professor of mathematics for thirty years at the University of Bologna [2]. Unfortunately, he did not pub- lish his discoveries so he wasn't given too much credit. He was able to solve the equation: X3 + PX = Q This form is known as the depressed cubic form where P and Q are coefficients. In this case the coefficient of the quadratic or X2 term is 0. All cubic equations can be reduced to this form, but mathematicians during this time were only fa- miliar with the positive solutions. During these times, mathematicians assumed both P and Q to be positive num- bers. The formula that Del Ferro obtained only works using this assumption. His method works with cubic equations only in this depressed form. Del Ferro's formula can be given in the following form: Equation: X3 + PX = Q Q2 P 3 Discriminant: ∆ = 4 + 27 qp qp 3 Q 3 Q Root: X = ∆ + 2 − ∆ − 2 The discriminant of any polynomial is a quantity that depends on the coeffi- cients and determines various properties of the roots. For both quadratic and 6 Published by DigitalCommons@SHU, 2021 7 Academic Festival, Event 31 [2021] cubic equations, the roots are real and distinct if the discriminant is positive, they are real with at least two equal if the discriminant is equal to zero, and they include a conjugate pair of complex roots if the discriminant is negative. After Del Ferro discovered his solution, another Italian mathematician known as Gerolamo Cardano made significant contributions as well. Cardano lived from 1501 to 1576 and was a Renaissance scholar and physician as well as a math- ematician [2]. He is famous as the author of ArsMagna. This is a book that marked the beginnings of the subject of modern algebra. In this book he de- scribes Del Ferro's formula. Cardano was the first to actually publish the solu- tion to the cubic equation. The difference is, that Cardano made the solution work for values of P amd Q which did not necessarily have to be positive. It is valid irrespective of the signs of P and Q. Cardano's formula is as follows: q p q p 3 Q 3 Q Root: X = 2 + ∆ − 2 − ∆ Formally, it is the same as del Ferro's formula, except for the + sign in the sec- ond term which can be explained by reversing the sign of the expression in the second cube root in del Ferro's formula [2]. After the cubic solution had been presented, the next main focus would be on the biquadratic or quartic solution.