The Abakus - the earliest computing machine. Is manually operated, and uses decimal count to store state, meaning every time the counter reaches 10 a 1 bid is move right and the decimal counter resets, thus holding state of count.

The step reckoner - The step reckoner was a digital mechanical calculator invented by the German mathematician Gottfried Wilhelm Leibniz around 1672.
Its basic function is to count, adding up & subtraction (similar to abakus), yet can also do multilications and devide, and could do all four of this operations automaticaly.

Charles Babage and the Difference Engine - much more complex machine that could approximate polynomials.

Plynomials:
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. An example of a polynomial of a single indeterminate, x, is x2 − 4x + 7. An example in three variables is x3 + 2xyz2 − yz + 1.
Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated problems in the sciences; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, central concepts in algebra and algebraic geometry.

Charles Babage and the The Analytical Engine - was a proposed mechanical general-purpose computer designed by English mathematician and computer pioneer Charles Babbage.[2][3] It was first described in 1837 as the successor to Babbage's difference engine, a design for a mechanical computer.[4] The Analytical Engine incorporated an arithmetic logic unit, control flow in the form of conditional branching and loops, and integrated memory, making it the first design for a general-purpose computer that could be described in modern terms as Turing-complete.[5][6] In other words, the logical structure of the Analytical Engine was essentially the same as that which has dominated computer design in the electronic era.[3]

The tabulating machine - The tabulating machine was an electromechanical machine designed to assist in summarizing information stored on punched cards. Invented by Herman Hollerith, the machine was developed to help process data for the 1890 U.S. Census. Later models were widely used for business applications such as accounting and inventory control. It spawned a class of machines, known as unit record equipment, and the data processing industry.
was 10X faster than the analytical engine.

Electronic Computing

Harvard mark 1 - The IBM Automatic Sequence Controlled Calculator (ASCC), called Mark I by Harvard University’s staff,[1] was a general purpose electromechanical computer that was used in the war effort during the last part of World War II.

One of the first programs to run on the Mark I was initiated on 29 March 1944[2] by John von Neumann. At that time, von Neumann was working on the Manhattan project, and needed to determine whether implosion was a viable choice to detonate the atomic bomb that would be used a year later. The Mark I also computed and printed mathematical tables, which had been the initial goal of British inventor Charles Babbage for his "analytical engine".

Mechanical Relay - Mechanical relays are devices that can turn on or turn off the power supplied to another device, like a switch. However, instead of having a person flip the switch, mechanical relays switch when provided with a small amount of power. This allows high-power circuits to be controlled by low-power devices.
Thermionic valve - The type known as a thermionic tube or thermionic valve uses the phenomenon of thermionic emission of electrons from a heated cathode and is used for a number of fundamental electronic functions such as signal amplification and current rectification. Vacum tube.

Colossus mark 1 - Colossus was a set of computers developed by British codebreakers in the years 1943–1945 to help in the cryptanalysis of the Lorenz cipher. Colossus used thermionic valves (vacuum tubes) to perform Boolean and counting operations. Colossus is thus regarded[3] as the world's first programmable, electronic, digital computer, although it was programmed by switches and plugs and not by a stored program.

Transistors - A transistor is a semiconductor device used to amplify or switch electronic signals and electrical power. It is composed of semiconductor material usually with at least three terminals for connection to an external circuit. A voltage or current applied to one pair of the transistor's terminals controls the current through another pair of terminals. Because the controlled (output) power can be higher than the controlling (input) power, a transistor can amplify a signal. Today, some transistors are packaged individually, but many more are found embedded in integrated circuits.

Semiconductors - A semiconductor material has an electrical conductivity value falling between that of a metal, like copper, gold, etc. and an insulator, such as glass. Their resistancedecreases as their temperature increases, which is behaviour opposite to that of a metal. Their conducting properties may be altered in useful ways by the deliberate, controlled introduction of impurities ("doping") into the crystal structure. Where two differently-doped regions exist in the same crystal, a semiconductor junction is created. The behavior of charge carriers which include electrons, ions and electron holes at these junctions is the basis of diodes, transistors and all modern electronics. Some examples of semiconductors are silicon, germanium, and gallium arsenide. After silicon, gallium arsenide is the second most common semiconductor[citation needed] used in laser diodes, solar cells, microwave frequency integrated circuits, and others. Silicon is a critical element for fabricating most electronic circuits.

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How Computers Calculate: Half & Full Adders

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