In the 7th century, Indian mathematician Brahmagupta gave the first
explanation of the Hindu-Arabic numeral system and the use of zero as both
a placeholder and a decimal digit.

Approximately around the year 825, Persian mathematician Al-Khwarizmi wrote
a book, On the Calculation with Hindu Numerals, that was principally
responsible for the diffusion of the Indian system of numeration in the
Middle East and then Europe. Around the 12th century, there was translation
of this book written into Latin: Algoritmi de numero Indorum. These books
presented newer concepts to perform a series of steps in order to
accomplish a task such as the systematic application of arithmetic to
algebra. By derivation from his name, we have the term algorithm.

Around the 3rd century BCE, Indian mathematician Pingala invented the
binary numeral system. In this system, still used today to process all
modern computers, a sequence of ones and zeros can represent any number.

In 1703, Gottfried Leibniz developed logic in a formal, mathematical sense
with his writings on the binary numeral system. In his system, the ones and
zeros also represent true and false values or on and off states. But it
took more than a century before George Boole published his boolean algebra
in 1854 with a complete system that allowed computational processes to be
mathematically modelled.

By this time, the first mechanical devices driven by a binary pattern had
been invented. The industrial revolution had driven forward the
mechanization of many tasks, and this included weaving. Punch cards
controlled Joseph Marie Jacquard's loom in 1801, where a hole punched in
the card indicated a binary one and an unpunched spot indicated a binary
zero. Jacquard's loom was far from being a computer, but it did illustrate
that machines could be driven by binary systems.