We all use algebra. Even if it’s for the simple stuff, we use some form of algebra in our everyday lives. While reading chapters 1-10, I came across the word algebra and became quite curious about the subject for I have never really understood nor cared for it honestly, I just figured it’s the usual add, subtract and divide. I should’ve known it was more to it than that, especially I, being a math major and all of that. I remember first learning algebra and how hard I found it to be, later realizing how much it fun it was when I had finally grasped the concepts of it. It seemed complex at first because I did not have the inquisitive and wondering mind that I have now. After years of brain re-wiring I am now basically skeptical and want to see the logic behind everything. So, what is algebra? Who are or better yet, who is, the father of Algebra? Where did it come from? I might jokingly add, why must we learn it todays schooling system? Hopefully I can answer these burning questions within this paper.
Algebra didn’t really have a “name” until around 825 A.D when a mathematician by the name of Mohammed al- Khwarizmi who wrote the book called, Hidab al-jar wal-muqubala or, The Compendious Book on Calculation by Completion and Balancing in the area known as Baghdad in the middle east. From then on from the spread of “al-jabr” from the middle east into Europe, we have the processes of modernizing the name to give us what we know today as algebra. I first stated that algebra didn’t really have a name until 825 A.D and that’s because the first instances of algebra are found in Egypt and Babylon. Some would even say the origins of Algebra hails from these two known countries around the 1st millennium B.C. Even though they are both credited as places of origin for Algebra, the Babylonians use of algebra and the Egyptian use of algebra varied. It varied to the point that Babylonians thought the Egyptians use of algebra was too elementary and sought to create and apply a higher level of mathematics.
The variances between Egyptians algebra and the Babylonian algebra was their numerical systems. The Babylonians numerical system was positional but also sexagesimal. A positional numeral system is a system in which the value of a numeral symbol depends on its position. They used ones and tens symbols to represent the numbers 1 through 59. The word sexagesimal, when broken down comes the Latin word sexagesimus which means “sixtieth” or something relating to the number sixty. Hence when scholars say they had a sexagesimal numeral system, scholars mean they operated on a numeral system called “base 60”. This system was passed down from the Sumerians around the 3rd millennium. From this numerical system, they had tables of products, conversion with coefficients and fractions which they used in their everyday life. One would believe the reason why they were so detailed and sought to apply a higher level of mathematics was because they used it every day. No doubt, ancient Babylon was advanced society wise, they had concepts known as civil servants and even a pretty complex yet efficient government and in a way, math was its own identity.
When it comes to Egypt, many scholar disagree about a ton of stuff such as the nature of Egyptian mathematics, that being because ancient Egyptians wrote on papyrus with ink, which if we’re talking long term doesn’t last. Egypt used two forms of writing. For more practical reasons like accounting and civil services, they used ink on papyrus and for mathematics that tied in with their history they wrote in hieroglyphics. This being another difference between the Egyptians and Babylonians, the preservation of their knowledge. Babylonians used clay tablets which many were found. So, in turn there’s not many evidence aside from the periods and the calendars to vouch for Egyptian mathematics. As I’ve stated before, most Babylonians thought Egyptian mathematics were too simple, because their mathematics was not simple, in fact it was practical. Egyptians had to be. For example, Egypt being by the Nile had to have great irrigation systems because the Nile river routinely flooded. The Egyptian numerical system, used from 3000 BC to the first millennium, was based on the scale of 10. It was often rounded up, but they unlike the Babylonians, had no concepts of positional numerals, it was often unordered.
Aside from their numeral systems the way they went about their calculations was different as well. Babylonians used algorithmic calculations to solve problems that would today be solved by quadratic and linear equations. Babylonian algebra had its own set of distinctions such as:
- They had no zero (if they had no zero, makes me wonder if they thought the concept of “nothing” doesn’t exist. Or if they were practicing with numerals like the modern set of Natural numbers).
- Circular measurement
- Square roots
- Cubic equations, sometimes even equations with several unknowns
- They used the Pythagorean theorem
- Quadratic formula
- Modeling exponential growth and doubling time.
For example, when it came to quadratic or cubic equations they used the equations in the form of
x2+bx=c for quadratic and ax3+bx3=c for cubic.
The Egyptians as stated were that advanced as the Babylonians. They only used addition and subtraction do what needed to be done. For multiplication, they used the doubling system. For example, if they needed to know what 12×2 was, they would just add 12+12 to get 24, which is correct. As for fractions, they used the sum of distinct fractions, for example ¾= ½ + ¼. Even though the Egyptians didn’t have the knowledge for circular measurement didn’t mean they couldn’t do it. They did have to find a way to do so because again, the irrigations systems needed to be top notch because of annual flooding of the Nile River. For Example, let’s say they needed to find the area of a circle with a diameter of 9. They noticed the number 9 was very close to the number 8. Hence, they could just multiple 9 x (8/9) and then square it, giving a margin of error of less than one percent.
I keep going on and on about the irrigation systems and how it ties into the Nile river’s annual flooding that I keep forgetting about the calendars and how they used mathematics to figure out the position of the stars to create yearly concepts like, “new year’s” and things of that sort. The Egyptians used mathematics to look and create times while measuring the rise and fall of the Nile. They started their day at Sunrise and divide the day into 12 hours. Although it was accurate, it was accurate enough to allow them to avoid, again, the flooding in return allowed them to grow crops and created great sustention for many years.
The Egyptians and the Babylonian math had many differences but they also had similarities like the concept of practicality and using their mathematical skills to further better their societies. This helps me reflect on how universal modern math is. Whether it be theoretical or applied, math can literally be used in any field, even the arts, many artists were known to be interested in mathematics as well. So jokingly, I interject that mathematicians are artists as well. Let’s get to the main reason why I’m even typing this essay: Mohammed al- Khwarizmi.
Mohammed al- Khwarizmi was a Persian mathematician who revolutionized algebra with his book, Hidab al-jabr al-muqubala in which he uses Hindu-Arabic numerals. Because of his origins, modern day- Uzbekistan, it’s only natural he be influenced by the Babylonians and the Indians. The name represents the two operations he shares in his book. Al-jabar is Arabic for restoration and al-muqabala means balancing, therefore translating the title of the book to, The Compendious Book on Calculation by Completion and Balancing. From this book comes the concept of adding and subjecting, and the transfer of one term from either side of an equation, hence the balancing part. It starts with the bringing forth of an idea of equations to the first and second degree. Then towards the end, begins the application of such ideas.
His book, The Compendious Book on Calculation by Completion and Balancing, gaining wide spread recognition help spread the use of Hindu-Arabic numerals and is now the most use symbolic representation for numbers in the world. Like his predecessor’s numeral system, the Hindu-Arabic numeral system is designed to be positional. In other words, the introduction of this numeral systems brought about a new way to count. The reason why it was so revolutionary is because, during his time, there was no way to use notations such as x or y, because they hadn’t been invented yet. Because there wasn’t a way to pair the numbers with x,y notations, he didn’t have access to a tool that could’ve helped him explain his ideas a little bit better. So, in other words (pun intended), he had to explain his ideas with…. words. The irony is what gets me, but also makes me wonder about other things such as: how did he get his points across? Logic and proofs?
In relation to me as a student creating proofs are already hard enough even with the tools that I have now such as notations, numerals, and definitions so I cannot imagine having to prove or explain something can be done without such tools. According to History of mathematics by Carl B. Boyer, Arabs at that time, loved good and clear arguments as well as systematic organizations. They were perceived to be level headed and down to earth when it came to mathematics. That would probably explain how easy it was for Khwarizmi to explain himself when it came to his books. In his book he states, the reason for writing the book was to teach what was easiest. From measurement of lands, to geometrical computations and digging of canals, to trade and lawsuits, he touched on in his book. As I’ve stated before, mathematics is universal and be used for literally anything if we sat and thought about it, so it’s only natural that one of the leading pioneers of mathematics realized this as well. Usually, when it comes to how diverse mathematics truly its, it is only stated and not really shown until one does deep research about a subject. Did you know law has mathematics? Such as criminal law, there are (excuse my example) different grams of weed or marijuana that you can walk away free with? If you’re caught with less than a gram of weed, you’ll get a warning and, maybe a class, but other than that, you are free to walk. This is only in the state of Texas I’m not sure about other states but it’s around the same I’ve heard. Amazing that math no matter how small is included in something such as criminal law.
Of course, when it comes to who really is the father of algebra the world of mathematics has two contenders: Khwarizmi and Diophantus. Diophantus was a Greek mathematician who also famous for his work in the field of algebra. Diophantus is known for two pieces of work, even though they are both incomplete due to his life circumstances. One is a small fragment on polygonal numbers and the second is called Arithmetica. Arthmetica probably established Diophantus in the 3rd century because it inspired a rebirth in number theory and was the first known work to employ modern algebra. The way they approach algebra is apparently different. Some claim Diophantus focused on the theory of numbers, which makes sense. When people claim, he employed “modern algebra” they’re probably talking about his use of negative and positive numbers. I personally think Khwarizmi deserves that title because of the way he approached the subject. He taught it in the elementary form and just for the sake of it. Plus, when we first learn algebra, we learn it in the elementary form. I’m not sure about others but I first tried my hands with algebra in 8th grade, and now looking back, it was simple.
When why algebra is taught in schools, one of the reasons stated is because it’s the foundation of advanced or higher up mathematics. It helps develop critical thinking skills and is, surprisingly, the first taste we mathematicians get of logic. Algebra has been around for a long time and I don’t think it’ll be dying out soon. In fact, I think it’ll advance and might incite a re-birth of algebra. It’ll build on linear algebra, abstract algebra and many other forms of algebra and I am really excited to see where it leads.