Short answers, pulled from the story.
The Ishango bone, discovered near the headwaters of the Nile river in northeastern Congo, stands as the oldest known artifact suggesting mathematical thought, dating back more than 20,000 years. This piece of bone features a series of marks carved in three columns that run the length of the object, leading scholars to debate whether it represents a tally of prime numbers or a six-month lunar calendar.
The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the first civilization in Mesopotamia and developed a complex system of metrology from 3000 BC. By 2500 BC, Sumerian scribes were writing multiplication tables on clay tablets and solving geometric exercises and division problems.
This era of mathematical rigor began with Thales of Miletus, who is credited with the first use of deductive reasoning applied to geometry, and Pythagoras of Samos, who established the Pythagorean School with the doctrine that mathematics ruled the universe. The study of mathematics as a demonstrative discipline truly began in the 6th century BC, transforming it from a collection of rules of thumb into a science.
The true revolution came in the 7th century when Brahmagupta lucidly explained the use of zero as both a placeholder and a decimal digit in the Brahma-sphuta-siddhanta. This concept was transmitted to the Islamic world and then to Europe, displacing all older number systems.
In the 9th century, the Persian mathematician Muhammad ibn Musa al-Khwarizmi wrote The Compendious Book on Calculation by Completion and Balancing, which introduced algebra as an elementary form for its own sake. The word algorithm is derived from the Latinization of his name, Algoritmi, and the word algebra comes from the title of his work, Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala.
Isaac Newton and Gottfried Wilhelm Leibniz independently developed infinitesimal calculus, bringing together the concepts of limits and derivatives to explain the laws of physics. This era also saw the correspondence of Pierre de Fermat and Blaise Pascal, which set the groundwork for probability theory and combinatorics.