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A microchip is a small integrated circuit (IC chip) that replaces a series of resistors, transistors, diodes, and other electronic components to perform some standard function. One common example is the 555 timing circuit. About a billion 555 timing microchips are produced each year.
The integrated circuit dates back to the late 1940s, about four years after WWII. It is easier and cheaper to print a common circuit in an integrated fashion than to build each one from individual components by hand. In addition, the "micro chip" takes up considerably less space. A hand built timer circuit might consume several square inches of breadboard, while the 555 chip is about 1/2 inch in size. Five years ago we were building circuits on the 100 nanometer scale (100 billionths of a meter). Now we're pushing down into tens of nanometers.
In 1961 when American launched Alan Shepard into space, the computing equipment filled about three rooms. Today all that same processing power could easily be handled by a single notebook PC.