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While Babbage’s designs were the first for a mechanical computer, it was never built until part of it was assembled a few years ago for a museum display. Also, its primary purpose was to calculate and accurately print mathematical tables, rather than to speed up calculations. So I think a question like this is not complete without mentioning some other candidates, and of these the American developed ENIAC is the first and foremost. It was the first electronic general purpose computer.
The main switching was carried out by some 18000 vacuum tubes (thermionic valves) and it had to be programmed by painstakingly connecting up hundreds of cables. It worked in decimal rather than binary. It was originally designed to calculate wartime ballistic trajectories but as the war was over by the time it was completed, it was put to other practical uses. It served until 1955.
Also worth mentioning is the experimental Manchester Baby, built at Manchester University in the UK. This was the first machine to have all the elements of a modern computer, including an electronically stored program. It first ran in 1948, and one of the first programs tested was to find the highest factor of 2^18, which it did in 52 minutes having completed 3.5 million operations. It even had a CRT display:
This machine led to a bigger, more practical version called Manchester Mark 1 in 1949, shown below:
The final production version was known as the Ferranti Mark 1. Delivered to the University in 1951, it became the world’s first commercially available general-purpose computer: 
This was followed shortly in the USA by UNIVAC, a commercial development of the ENIAC computer.
There were many other computers designed in the 1930s and 1940s which I have not mentioned, all of which helped contribute to the development of the modern computer we know today. For example, the Atanasoff-Berry computer was the first to have the modern architecture as proposed by Von Neuman, which included a system bus, a main memory, an ALU (arithmetic logic unit) and a control unit. It also had input and output (I/O) devices operated by a control unit.
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