I don't know about the calculator, but a gentleman called Charles Babbage is credited with inventing the first working computer back in the time of Queen Victoria. It was called the "analytical engine". A young lady friend of his, Ada Lovelace, who was the niece of Lord Byron was very enthusiastic about it and used to demonstrate it for him. The early computer programming language, ADA, was named after her. Despite its posh title and the fact that it has been called the first working computer, the analytical engine was probably little more than a sophisticated adding machine, i.e. Calculator.
Well technically the first calculator was not electric. As Felicity said Babbage made the first computer and therefore may be your answer. However, I would like to point out that the Chinese or the middle eastern civilization had counting aparati called, an abacus. It was a simple hand controlled device of beads, which could allow a practiced user to do addition, subtraction, multiplication and division rapidly.