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Origins of algebra

Where did the word "Algebra" and its underlying ideas come from? Created by Sal Khan.

Video transcript

What I want to do in this video is think about the origins of algebra. The origins of algebra, and the word, especially in association with the ideas that algebra now represents, comes from this book, or actually this is a page of the book right over there. The English translation for the title of this book is the "Compendious Book on Calculation by Completion and Balancing." And it was written by a Persian mathematician who lived in Baghdad in, I believe, it was in the eighth or ninth century. I believe it was actually 820 AD when he wrote this book. AD. And algebra is the Arabic word, that here is the actual title that he gave to it, which is the Arabic title. Algebra means restoration or completion. Restoration or completion. And he associated it in his book with a very specific operation, really taking something from one side of an equation to another side of an equation. But we can actually see it right over here, and I don't know Arabic, but I actually do know some languages that seems to have borrowed a little bit from Arabic, or maybe it went the other way around. But this says Al-kitab, and I know just enough Urdu and Hindi to understand a good India movie, but Al-kitab, kitab means book. So this part is book. Book. Al-mukhtasar, well, I think that means compendious, because I don't know the word for compendious and that seems like that. Fihisab, hisab means calculation in Hindi or Urdu, so this is calculation. Calculation. Al-gabr, this is the root. This is the famous algebra, this is where it shows up. So this is for completion, you could view that as completion. Completion. And then wa'l-muqabala, and that means essentially balancing. Balancing. Completion and balancing. So if we wanted to translate it-- I know this isn't a video on translating Arabic, but the book, I guess this is saying compendious on calculation by completion and balancing is the rough translation right over there. But that is the source of the word algebra, and this is a very, very, very important book. Not just because it was the first use of the word algebra, but many people viewed this book as the first time that algebra took a lot of its modern-- took on many of its modern ideas. Ideas of balancing an equation. The abstract problem itself, not trying to do one off problems here or there. But al-Khwarizmi was not the first person, and just to get an idea of where all this is happening. So he was hanging out in Baghdad, and this part of the world shows up a lot in the history of algebra. But he was hanging out right there in around the eighth or ninth century. So let me draw a time line here, just so we can appreciate everything. So that is timeline, and then whether or not you are religious, most of our modern dates are dependent on the birth of Jesus, so that is right there. Maybe I'll put a cross over there to signify that. When we want to be non-religious, we say the common era. Before the common era, when we want to be religious we say AD, which means in the year of our lord. I don't know the Latin, Anno Domini, I believe, year of our lord. And then when we want-- in the religious context, instead of saying before common era, we say before Christ, BC. But either way, so this is 1000 in the common era. This is 2000 in the common era. And obviously, we are sitting-- at least when I'm making this video, I'm sitting right about there. And then this is 1000 before the common era, and this is 2000 before the common era. So the first traces-- and I'm skipping out, and really, it's just what we can find. I'm sure if we were able to dig more, we might be able to find other evidence of different civilizations and different people stumbling on many of the ideas in algebra. But our first records of people really exploring the ideas that are hit upon in algebra come from ancient Babylon around 2000 years before the common era, before Christ. So right around there there are stone tablets where it looks like people were exploring some of the fundamental ideas of algebra. They weren't using the same symbols. They weren't using the same ways of representing the numbers, but it was algebra that they were working on. And that was, once again, in this part of the world. Babylon was right about there. And Babylon, it's kind of kept the tradition of Sumeria. This whole region was called Mesopotamia, Greek for between two rivers. But that's the first traces of people that we know of that where people were starting to do what we would call real, real algebra. And then you fast forward. And I'm sure we're missing-- and I'm sure even our historians don't know all of the different instances of people using algebra, but the major contributions to algebra, we saw it here in Babylon 2000 years ago. And then if we fast forward to about 200 to 300 AD, so right over there, you have a Greek gentleman who lived in Alexandria. So this is Greece right over here, but he lived in Alexandria, which at the time was part of the Roman Empire. So Alexandria is right over here, and he was a gentleman by the name of Diophantus, or Diophantus. I don't know how to pronounce it, Diophantus. And he is sometimes credited with being the father of algebra, and it's debatable whether it's Diophantus or al-Khwarizmi. al-Khwarizmi, who kind of started using these terms of balancing equations and talking about math in a purer way, while Diophantus was more focused on particular problems. And both of them were kind of beat to the punch by the Babylonians, although they all did contribute in their own way. It's not like they were just copying what the Babylonians did. They had their own unique contributions to what we now consider algebra. But many, especially Western historians, associate Diophantus as the father of algebra. And now, al-Khwarizmi is sometimes what other people would argue as the father of algebra, so he made significant contributions. And if you go to 600 AD-- so if you go to about 600 AD, another famous mathematician in the history of algebra was Brahmagupta, in India. Brahmagupta, in India. So obviously, and actually, I don't know where in India he lived. I should look that up, but roughly in that part of the world. And he also made significant contributions. And then you have al-Khwarizmi, who shows up right there, al-Khwarizmi. And he is the gentleman that definitely we credit with the name algebra, comes from Arabic for restoration, and some people also consider him to be, if not the father of algebra, although some people say he is the father, he is one of the fathers of algebra because he really started to think about algebra in the abstract sense, devoid of some specific problems and a lot of the way a modern mathematician would start to think about the field.