The Ancient Egyptians developed a system of numbers in the third millenium BC as did the Sumerians (of Modern Iraq). Both civilisations used versions of abacus. The Chinese developed their numbers about a thousand years later. In 876 BC, a symbol for zero was invented in India.
However, when we think of ancient mathematicians, most people remember the Greeks. This is because maths was central to their way of life. The Greeks used maths to find out more about the world around them. Pythagoras was not only a mathematician but a philosopher and religious leader.
After Diophantus of Alexandria wrote the first book on Algebra, in 210 AD, western mathematics went into decline for many years. Elsewhere, in India, the decimal system was developed in about 600 AD. The work of the Greeks was kept safe and enlarged upon by the Mohammedans of Northern Africa, who were far more tolerant to novel ways of thinking than the Christians of the time. However, what was not tolerated by Islam was pictures. In order to decorate their walls Muslims began to develop pattern. As their patterns became more intricate so did their maths. The greatest Mohammedan mathematician was Muhammad al-Khwarizmi who combined algebra and the decimal system.
At the beginning of the 13th Century the first signs of the Renaissance, still 200 years away, could be seen in the work of Italian mathematician Leonardo Fibonacci. Fibonacci studied rabbits breeding and noticed a pattern that can be traced throughout nature. Indeed Fibonacci’s series is related to the golden ratio (which the Ancient Greeks had discovered and used to design their temples and artefacts). During the Renaissance all these strands were pulled together and a chain of discoveries from trigonometry (1550) through logarithms (1614), probability theory (1654), the binary system and calculus (1679), and the first calculator (1822), to mention a few, have led directly to the production of the computer that I write this on today.