"Computers" have been around for a very, very long time. But the definition of what makes something a computer has changed a great deal. And the progress made on developing computers was made by many many people, not just one "inventor". There are many people out there who would say that the first "computer" was the abacus, invented in Asia about 5000 years ago. But somehow I doubt that this is what you're looking for, so let's look a little more recently.
As time went on, there were a number of special devices invented to help with things like tax collecting, taking the census, etc. At first, these were purely mechanical, but by the start of the twentieth century, they were run by steam.
The first of the "modern" computers was invented during World War II, in 1941 by a German engineer named Konrad Zuse. The computer was called the Z3 and was used to help design German airplanes and missiles. A couple years later, in 1943, the Allied forces developed a computer called Colossus to help decode German messages. But since the Z3 was developed by the side that lost the war and Colossus stayed a military secret for many years, these computers didn't contribute much to the ones that came next.
Independent of the Colossus project, the next computer was the Mark I, designed by Howard H. Aiken, an engineer working with Harvard and IBM. The Mark I was positively huge, taking up half of a football field, but it helped to create ballistic charts for the US Navy during the war. Shortly after this, though, came the Electronic Numerical Integrator and Computer (ENIAC), developed by John Presper Eckert and John W. Mauchly, working with the government and the University of Pennsylvania. ENIAC was a lot like the Mark I, except that it ran about 1000 times faster.
Moving along, there were other computers like EDVAC (1945), UNIVAC I (1951), etc. But all these computers had something in common with the older computers - they were designed for a specific purpose and couldn't really be used for anything else. They also all worked by using vacuum tubes, which is what made them take up so much space. The invention of the transistor in 1947 by John Bardeen, Walter Brattain, and William Shockley at Bell Labs made the big difference from here.
Using transistors, computers came around that could store memory and even run programs. Soon they even had computer languages so that people could change the programs run by the computer when they wanted to. After a while, the focus on computer research came to be on making them smaller, giving us the kinds of computers that we have today.
I believe that first computer invented was ENIAC, an all-purpose computer, used for calculating artillery firing tables for US Army Ballistic Research Laboratory.
Humans have been communicating with each other in many different ways for several hundred thousand years.The first digital computer was invented in the 1830s by Charles Babbage but never built. The first electronic digital computers were invented in the 1930s and built in the 1940s. Electronic digital computers did not become a marketed product until the 1950s.
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