An A-to-Z History of Mathematics:

Mathematics is the science of numbers. To be precise, the Merriam-Webster dictionary defines mathematics as, the science of numbers and their operations, interrelations, combinations, generalizations, abstractions and of space configurations and their structure, measurement, transformations, and generalizations.

There are several different branches of mathematical science, which include algebra, geometry, and calculus.

Mathematics is not an invention. Discoveries and laws of science are not considered inventions since inventions are material things and processes. However, there is a history of mathematics, a relationship between mathematics and inventions and mathematical instruments themselves are considered inventions.

According to the book Mathematical Thought from Ancient to Modern Times, mathematics as an organized science did not exist until the classical Greek period from 600 to 300 BC. There were, however, prior civilizations in which the beginnings or rudiments of mathematics were formed.

For example, when civilization began to trade, a need to count was created. When humans traded goods, they needed a way to count the goods and to calculate the cost of those goods. The very first device for counting numbers was, of course, the human hand and in which fingers represented quantities. And to count beyond ten fingers, mankind used natural markers, rocks or shells.

From that point, tools such as counting boards and the abacus were invented. 

Here's a quick tally of important developments introduced throughout the ages, beginning from A to Z. 

ABACUS

One of the first tools for counting invented, the abacus was invented around 1200 B.C.E. in China and was used in many Ancient civilizations, including Persia and Egypt.

ACCOUNTING

The innovative Italians of the Renaissance (fourteenth through sixteenth century) are widely acknowledged to be the fathers of modern accounting.

ALGEBRA

The first treatise on algebra was written by Diophantus of Alexandria in the 3rd century B.C.E. Algebra comes from the Arabic word al-jabr, an ancient medical term meaning "the reunion of broken parts.'' Al-Khawarizmi is another early algebra scholar and was the first to teach the formal discipline. 

ARCHIMEDES

Archimedes was a mathematician and inventor from ancient Greece best known for his discovery of the relationship between the surface and volume of a sphere and its circumscribing cylinder for his formulation of a hydrostatic principle (Archimedes' principle) and for inventing the Archimedes screw (a device for raising water).

DIFFERENTIAL

Gottfried Wilhelm Leibniz (1646-1716) was a German philosopher, mathematician, and logician who is probably most well known for having invented differential and integral calculus. He did this independently of Sir Isaac Newton.

GRAPH

A graph is a pictorial representation of statistical data or of a functional relationship between variables. William Playfair (1759-1823) is generally viewed as the inventor of most graphical forms used to display data, including line plots, the bar chart, and the pie chart.

MATH SYMBOL

In 1557, the "=" sign was first used by Robert Record. In 1631, came the  ">" sign.

PYTHAGOREANISM

Pythagoreanism is a school of philosophy and a religious brotherhood believed to have been founded by Pythagoras of Samos, who settled in Croton in southern Italy about 525 BC. The group had a profound effect on the development of mathematics.

PROTRACTOR

The simple protractor is an ancient device. As an instrument used to construct and measure plane angles, the simple protractor looks like a semicircular disk marked with degrees, beginning with 0º to 180º.

The first complex protractor was created for plotting the position of a boat on navigational charts. Called a three-arm protractor or station pointer, it was invented in 1801 by Joseph Huddart, a U.S. naval captain. The center arm is fixed, while the outer two are rotatable and capable of being set at any angle relative to the center one.

SLIDE RULERS

Circular and rectangular slide rules, an instrument used for mathematical calculations, were both invented by mathematician William Oughtred.

ZERO

Zero was invented by the Hindu mathematicians Aryabhata and Varamihara in India around or shortly after the year 520 A.D.