Despite fevered fetishes of American exceptionalists, what exists now has a foundation on what came before, and much of what we consider "modern" came from Baghdad. Ignorant savages indeed.
[The story begins in AD830. The place: The House of Wisdom, Baghdad, Persia in the Islamic Golden Age. Works from around the world were brought to Baghdad for translation. Scholars of Babylonian astronomy, Indian numbers and Greek mathematics all came to the richest city in the world to embrace its powerful intellectual culture. A paper mill had been built in the city, and due to this minor industrial revolution, there was a proliferation of books and libraries.
Mohammad Bin Musa Al-Khwarizmi was the father of algebra. He introduced the idea of the zero (brought from India) as a place-holder in arithmetic, which give our calculations their columns of hundreds, tens and units. He also wrote the book that probably changed our world as much as any other: Hisab al-jabr wa al-muqabala, which translates as "The Compendious Book on Calculation by Completion and Balancing". From al-jabr came the word "algebra".
The of al-jabr progressed in the East, but it took hundreds of years for the scholars of western Europe to advance. Most notable of those who took the Lady Algebra to her next incarnation was Leonardo of Pisa, more commonly known as Fibonacci (c1170-1250).
Fibonacci, the most talented western mathematician of the Middle Ages, spread throughout Europe the use of the Hindu-Arabic numeral system, the system we use today. He also showed that new methods could be used to find solutions of some cubic equations, something that many students still learn, but which, at the time, was a stupendous advance.
The essence of algebra is solving or "reducing" equations to their simplest form. Taking away the confusion and the noise, and making sense of it all. To do that, we have to find specific methods. These methods are known as algorithms – the Latin and English corruption of al-Khwarizmi's name. You can try to solve algebraic equations in different ways. Guessing at solutions, for instance, is one way. It is possible, if you try for long enough, to guess the exact solution of a simple equation. Another way to find a solution is by means of geometric construction. The Greeks favoured this. But it is the invention of further algorithms that has driven, and will continue to drive forward, the language of algebra.]