SDS 460: The History of Algebra

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All right. For those of you who listen to FiveMinuteFriday with some regularity, you’ll know that I’m fond of digging into the history of various data science concepts, and few concepts stretched back further in history than algebra. All right. So first of all, what is algebra? Well, if we think of arithmetic as a mathematical field where we add and subtract, divide, multiply, it’s the simplest way of thinking about mathematics. Well, then algebra is arithmetic that includes non-numerical entities. So you have variables that are often expressed as a letter like X or Y, and this allows us to solve problems. So if we have algebra, we can then take a equation like 2X plus five equals 25. And we can rearrange the equation using algebraic tricks, like moving one number over to the other side of the equation, dividing both sides of the equation by a number. That allows us to isolate our variable. So we can find then that X in the equation, 2X plus five equals 25, is equal to 10. 

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So algebra today is hugely powerful. It’s used throughout data science and machine learning. Some examples of contemporary applications of algebra today include, just like we did with that very simple equation that I just talked you through, we can use it to solve for the unknowns in machine learning algorithms. And this can be some of the relatively simple machine learning algorithms like regression models, all the way through to the most sophisticated ones, like deep learning models. We can use algebra to reduce dimensionality. For example, there’s a technique called principle component analysis that allows us to distill a large number of dimensions down to a smaller, more manageable set that tend to be the most important set. 

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We can use linear algebra to rank search results. So we could rank webpages in order of importance using techniques like an eigenvector. We can get recommender systems, for example, to recommend movies to us on popular services like Netflix. And this is powered by linear algebra techniques like singular value decomposition. Then when we’re processing natural language, so handling my voice being a broadcast into your ears right now, or words on a page that are written somewhere, when we analyze that natural language, we use linear algebra techniques a lot like the singular value decomposition that I just mentioned, as well as something called matrix factorization. So a huge number of applications of linear algebra today at scale in machines and probably behind countless interactions that you have with machines every single day. So it’s ubiquitous today, but where did it come from?

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Well, first I’m going to tell you about the name, which is fascinating in itself. So the person responsible for naming algebra is someone named Abu Ja’far Muhammad ibn Musa, who lived from approximately 780 to 850. He was a medieval Persian mathematician, and he was often called al-Khwārizmī in Arabic, which means the man from Khwarazm, which is a city, a former center of Persian culture that is today in modern Uzbekistan. So this name, al-Khwārizmī, the man from Khwarazm, that can be pronounced in English is algorithm. Al-Khwārizmī, algorithm. So there’s a fun fact in and of itself that this guy is the basis of the word algorithm, but we’re not going to get dragged down that route. We’re going to stick on algebra for now. 

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So al-Khwārizmī, all those centuries ago, more than a millennium ago, wrote a book called The Compendious Book on Calculation by Completion and Balancing. So he wrote that book probably around the year 830. And some historians, for example, the gentleman named Victor Katz regard this book as the first true algebra text that is still in existence. What’s really interesting about this book name is it has the words completion and balancing in it, The Compendious Book on Calculation by Completion and Balancing. Well, that completion and balancing, those are the things that we do when we’re rearranging an equation to solve for X, for example.

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So when we move a term from one side of the equation to the other, that is completion. When we remove equivalent terms on both sides of the equation, that is balancing. But that first word in The Compendious Book on Calculation by Completion and Balancing, that first word, completion, is in Arabic pronounced algebra. So there’s the etymology of the word algebra and al-Khwārizmī. It was around quite a long time ago, writing books and defining whole fields like algebra. So the year 800, that sounds like a very long time ago, but algebra is actually much older than that. 

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So, while the medieval Islamic Arab empire made most of the significant early contributions in the modern symbolic algebra we use today, symbolic meaning the use of terms like X instead of just numbers, other cultures developed their own approaches to algebra much earlier. Babylonians have the earliest known style of algebra, and this was a rhetorical algebra, a spoken algebra. And this dates to as early as 1900 BC, so 4,000 years ago. The Egyptians had linear algebra dating back to 1650 BC. And we know that from papyruses, [papyre 00:07:12]. Egyptian documents from almost 4,000 years ago. There are Indian mathematical documents from around the sixth century before the common era with linear equations in them. The Greeks had a geometric algebra, so around 400 to 300 BC in the time of Plato. So there’s a famous book by Euclid called Elements. And in that book, there are lots of examples of this geometric algebra. So this shape-based type of algebra. 

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There’s another example of a culture that had algebra, the Chinese had linear equations solved in a book dating from 250 BC in a book called Nine Chapters on the Mathematical Art. Europeans got started much, much later on algebra in the late middle ages. And they weren’t starting from scratch. They were starting from the Arabic work on it. So in the 12th century, translated Arabic texts to Latin from the original Arabic. And by the 13th century, European mathematics began to rival the mathematics of other cultures that started much earlier. By the 15th century, European cultures became the primary carriers of the algebra Bhutan as there began to be much less work on mathematics in the Islamic empire around that time. And it’s that tradition from the European 15th century that has a continuous lineage to the algebra that we have today, including the algebra behind all of the machine learning algorithms that are powering the devices, the technology, that are being used by billions of people around the world today. 

(09:15):
Pretty cool, right? Hopefully you found that history of algebra interesting. I certainly do. I talk about the history of various mathematical disciplines, algebra, calculus, probability and statistics, and so on in my Machine Learning Foundations series of tutorials. That’s available for free on YouTube. I have a YouTube playlist called Machine Learning Foundations. Or you can check out the Udemy version of that content, which I’ve released in partnership with SuperDataScience, and is available usually for a very low price. It’s the same as what I have in YouTube, except that I have fully worked solutions to any exercises available in those Udemy videos. 

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Anyway, the purpose of today’s FiveMinuteFriday was not for commercial purposes whatsoever. I simply wanted to tell you about something that I find really interesting. We’ll come back with a future FiveMinuteFriday at some point and do histories of other mathematical areas like calculus, probability and statistics. All right. And one final thing is a quick announcement, that starting with episode 465, which will be in two weeks time, we will begin releasing guest episodes on Tuesday mornings, New York time. So historically we’ve released the SuperDataScience guest episodes on Wednesday evenings, but by releasing 36 hours earlier, we’ll be giving you two more morning commutes in your week to enjoy the episode. I can’t imagine any downsides to this change, but I didn’t want to catch you off guard when it happens. All right. Catch you on another episode soon.