Even before the first writing systems reached full development, counting systems and ways to record calculations were in common use. Counting and calculation by gathered stones or notches cut in wood soon evolved into more complex systems such as inscribing number symbols in clay.
On this Cuneiform Tablet is an example of the first known system of writing (and the first known written system of counting) can be seen. Cuneiform is the name of the writing system first developed in Babylonia around 3500 B.C. It spread throughout the region in the following millennia and was used to represent many different languages.
This babylonian and Egyptian mathematics site has information on early number systems and some good illustrations.
For an alternate view take a look at A brief outline of the history of Chinese mathematics.
An overview of the history of mathematics gives you a short summary of the growth of counting and mathematics with links to more detailed information.
The Abacus is a classic example of an early calculating device. The amazing thing about the Abacus is that it is still in common use today in many parts of the world.
Here is a good picture of an Abacus
The Abacus: The Art of Calculating with Beads is a fascinating site with a working Abacus (a java Applet) that you can play with.
The ancient Romans created an empire around the Mediterranean Sea from about 200 bc to AD 500. Their counting system was in common use in Europe until about 1000 years ago. The Roman system of writing numbers was replaced with the Arabic numbers (1,2,3,4,5,6,7,8,9…).
Here is a summary of how the system works
The term algorithm derives from ....!!!!!!!!!!!!
...", Kitab al-jabr wa al-muqabalah ("The Book of Integration and Equation"), was translated into Latin in the 12th century and originated the term algebra."
"Another work on Hindu-Arabic numerals is preserved only in a Latin translation, Algoritmi de numero Indorum ("Al-Khwarizmi Concerning the Hindu Art of Reckoning"). From the title originated the term algorithm" ... britannica.com
Mechanical calculators and aids to calculation