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What an interesting question! I think there are two big parts that need discussion: 1) What is Algebra? Did Al-Khwarizmi invent it? And 2) What changed with the publication of Al-Khwarizmi’s book الكتاب المختصر في حساب الجبر والمقابلة Al-kitāb al-mukhtaṣar fī ḥisāb al-ğabr wa’l-muqābala, The Compendious Book on Calculation by Completion and Balancing.

To answer in short: 1) It’s very complicated and 2) Not much, because his work was more of writing down things probably already known.

Let’s look at the first question. This is an extremely contentious question. In modern mathematics, algebra is one of two major approaches to math, along with Analysis, and is often said to be something along the lines “the study of the mathematical structure of objects” which on the outside looks rather like all of math. It is often used synonymously with the phrase “abstract algebra”, which is the field studying groups, rings, fields and all that come out of them (which is really a lot) and since the mid twentieth century has been one of the main fields of mathematics research. And I should say that I think this is a rather bad definition, I’ve actually never found a definition I really like even after doing an undergraduate degree in math.

At the grade school level, people generally mean the study of equations and variables, and particularly solving equation, which is only a small part of what a mathematician would mean. So when we are looking at the past, what do we mean by algebra?

This is where we want to start thinking about Al-Khwarizmi. Because the very name algebra comes from him, as you might well know. That very long Arabic title I had above? Take a look at the word “al-ğabr” (you may have also seen it as “al-jabr”, just pulled the transliteration from wiki as I’m too lazy to go and type the Arabic myself). That word would become (through Latin) the word “algebra” in modern English and Al-Khwarizmi is generally seen as the reason for that (why exactly, I’ll try and answer later). So what does Al-Khwarizmi mean? He uses it as part of the phrase which can be translated (as wiki does above) as “Completion and Balancing” or in my preferred translation by Roshdi Rashed as “restoration and compensation”. What does he mean by that? I don’t think it’s totally clear, or at least that’s what I’ve read. But if we had to take a stab at a guess, following Sesiano, I’d suggest that restoration is doing things like adding to both sides.

So if Al-Khwarizmi is using al-jabr in a technical sense, why does it come to be used for such a broad concept? To answer that I think we need to really look at the second question I floated above, What changed with the publication of The Compendious Book?

Well, to answer that question, let’s look at what Al-Khwarizmi says in the preface. I’ll use this old translation: https://archive.org/details/algebraofmohamme00khuwrich, because it’s free on the internet, but Rashed’s is more modern and I think prefferable, if you can find yourself a copy. So, what does Al-Khwarizmi say?

The learned in times which have passed away, and among nations which have ceased to exist, were constantly employed in writing books on the several departments of science and on the various branches of knowledge, bearing in mind those that were to come after them, and hoping for a reward proportionate to their ability, and trusting that their endeavours would meet with acknowledgment, attention, and remem- brance—content as they were even with a small degree of praise; small, if compared with the pains which they had undergone, and the difficulties which they had encountered in revealing the secrets and obscurities of science.

Some applied themselves to obtain information which was not known before them, and left it to posterity others commented upon the difficulties in the works left by their predecessors, and defined the best method (of study), or rendered the access (to science) easier or placed it more within reach ; others again discovered mistakes in preceding works, and arranged that which was confused, or adjusted what was irregular, and corrected the faults of their fellow-labourers, without arrogance towards them, or taking pride in what they did themselves.

That fondness for science, by which God has distinguished the Imam al Mamun, the Commander of the Faithful (besides the caliph which He has vouchsafed unto him by lawful succession, in the robe of which He has invested him, and with the honours of which He has adorned him), that affability and condescension which he shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on Calculating by (the rules of) Completion and Reduction, confining it to what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, law-suits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computation, and other objects of various sorts and kinds are concerned relying on the goodness of my intention therein, and hoping that the learned will reward it, by obtaining (for me) through their prayers the excellence of the Divine mercy in requital of which, may the choicest blessings and the abundant bounty of God be theirs ! My confidence rests with God, in this as in everything, and in Him I put my trust. He is the Lord of the Sublime Throne. May His blessing descend upon all the prophets and heavenly messengers !