In my unpublished manuscript, Jefferson Must Be Turning in His Grave, I address the Islamic "Golden Age." Islamic scholars have often referred to that period from the eighth century to the fifteenth century as their Golden Age. Ironically, while the Muslims were experiencing explosive growth in knowledge, European countries were going through the dark ages where the Catholic Church ruthlessly controlled scientific knowledge (remember, the Church accused Galileo of heresy for announcing the Earth revolves around the sun?). Some historians say it was the most productive time of all humanity. The critical point is the tolerance and co-existence of Muslim, Christian, and Jewish scholars under an Islamic caliphate. Toleration and cooperation between religions resulted in significant progress in medicine, mathematics, philosophy, law, architecture, and commerce.
The story here is how modern mathematics emerged from an ancient Islamic library. In her article "How modern mathematics emerged from a lost Islamic library," Adrienne Bernhard documents the almost extinct trail of how mathematics emerged to become the science that it is.
The story picks up in the eighth century in Baghdad, Iraq, one of the world's largest cities at that time. The Caliph Al-Rashid established a private collection of books, which later turned into a library which later converted to a public academy. During that time, it became an unrivaled center for the study of humanities and the sciences. If you need a mental image, think of the library at Hogwarts. One significant event and two established scholars initiated the overall shift to the math we know of today.
First, the numbers we recognize originated from Arabic script. While Hindu mathematicians developed the binary system sometime in the 5th century, Muslim scholars formalized the concept with Arabic scripture. It was interesting that dual systems of numbers consisting of Roman Numerals and Arabic script competed for recognition for several hundred years. The Arabic script won out as it was easier to use for mathematical and astronomical calculations. Using roman numerals for tabulations is very cumbersome. Try multiplying MMXXI by LVII. In the tenth century, Muslims introduced the Arabic script into Europe, and the fall of the roman numeral empire started precipitously. While most Americans and Europeans are versed in roman numerals, they are not in everyday use. You still see them used on some coins, the copyright date on films and tv programs, and some analog clocks to show the time. The National Football League still uses Roman numerals to designate the Super Bowl.
Two world-class scholars studied at this institution. First, there was a ninth-century Persian, Al-Khwarizmi, who developed a systematic way of solving quadratic equations. Those not familiar with such algorithms should look at this: Ax2 + Bx + C = 0. In the formulae, you solve for x. Some people refer to this as algebra, which comes from the Arabic word al-jabr, which means "the restoring of broken parts."
The second scholar of note came almost 400 years later. He was an Italian thirteenth-century mathematician known as Fibonacci. While Fibonacci discovered a sequence of numbering that shows up frequently in the natural world, his most extraordinary claim was a book published in the West encapsulating the Hindu-Arabic numeric system, al-Khwarizmi's quadratic equations, and other formulas to be used in the transaction of money.
In the middle of the thirteenth-century, the Mongols invaded the Middle East and in 1258 captured Baghdad and destroyed the Caliph's library. According to legend, the Mongols tossed so many manuscripts into the Tigris River that its waters turned black from the ink. No trace remains of this library, but Fibonacci's publications rescued some of the knowledge of mathematics. As Europe was entering its Renaissance, this knowledge helped fuel that change that made the European countries a bastion of scientific knowledge. Isaac Newton, in 1675, is quoted as writing, "If I have seen further, it is by standing on the shoulders of Giants." He uses that metaphor to indicate that the significant thinkers before his time had contributed knowledge for him to make even more intellectual progress. It would be safe to say that many of those giants were scholars in Caliph Al-Rashid's library.