Our digital numeral system came first from India, but spent a long time as the dominant system in the Arabic kingdoms before entering Latin and Europe. The numbers aren’t really Arabic in the way a keffiyeh is; but they became Arabic by adoption.
The Jain sect in India believed that numbers were divine; I don’t know much about their work, which may have been going on since BC years. We know that by the time Arab invaders took Sanskrit mathematics books to Baghdad for translation, the concepts of place value and equations were established and explored. Most writing systems of the time used their alphabetic letters to stand for numbers, or else they used some form of a tally-mark system. The Indian system used only 10 symbols and, significantly, included zero. I wonder sometimes if kids in first and second grade would find their work more interesting if it was presented as it came in history: a stunning breakthrough that had taken mankind several millenia to produce, and changed the world ever after.
The figures in early Sanskrit books are similar to our numbers, but not the same. They’re similar in that they aren’t letters and they aren’t based on tally marks. You can see a table of their evolution here.
Sanskrit books referred to solving equations as “pulverizing” them: you started with a lot of stuff that had to be ground down by processes, like crushing grain or sanding wood. At the end, the finished product stood alone: a number, instead of a sculpture or an iron nugget. When these books were translated into Arabic at the Baghdad House of Wisdom (late 700s), Persian astronomers picked up the system quickly. Al-Khwarizmi is the most famous mathematician of the new, early system. He mastered the Indian ideas and wrote treatises to explain how to use them. When his name was translated into Latin much later, it came out as Algorithmus. His treatise was titled Al-Jabr, The Transformations. This title went into Latin as algebra. But nobody translated Al-Jabr into Latin for a long time.
Persian and Arabic scholars worked with the ideas for a long time in isolation. The next major treatise was by Omar Khayyam in the 11th century. His work defined arithmetic and algebra as separate mathematical skills. Algebra, he wrote, was the use of equations to find unknown numbers with polynomials. Further, he wrote about irrational numbers and explained the mathematics of conic sections.
It’s not clear at what point negative numbers were recognized. They’re not in Al-Khwarizmi’s book, and I’m not sure about Khayyam’s. At any rate, a great deal of complex mathematics was in experimental stages in Baghdad and Alexandria, insulated from Europe by the Arabic language. Arabic merchants were now using the digit system for everyday calculation. There were two key points when Europeans entered the Arabic world, learned the number system, and brought it back into Latin. Muslim Spain was the first point of contact, in the 900s. (more later)