The first person to write about them was Gerolamo Cardano in 1545, but he doesn't seem to have taken them seriously.
See http://en.wikipedia.org/wiki/Gerolamo_Cardano .
The first serious use of imaginary numbers (better, complex numbers) was by Rafael Bombelli, published in 1572. He used them as intermediate steps when solving cubic equations.
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Rene Descartes came up with the word imaginary in 1637 to describe them. It was a derogatory term. He (and many other mathematicians of that age) did not like imaginary numbers. Many people didn't believe in them, because they were not real.
It has been known from very ancient times that the length of the diagonal of a unit square is not a rational number. There were no specific mathematicians who "discovered" real numbers. Furthermore, all mathematicians of any significance, contributed to our understanding of real numbers.
The 16th century Italian mathematician, Gerolamo Cardano was the first to use imaginary and complex numbers in his work on cubic equations.
In mathematics, an imaginary number is a number whose square is a negative real number and written in the form bi where i is the imaginary number √(-1) and b is real.A complex number is a number with both real and imaginary numbers, such as (3+2i), where 3 is real and 2i is imaginary.Imaginary numbers were 'invented' by Gerolamo Cardano in the 1500's while solving cubic and quartic equations although it is said he did not understand their properties, and they were not properly defined until 1572 by Rafael Bombelli, although he did not name them imaginary numbers.The name came from Descartes in his book "La Geometrie" where it was meant to be derogatory and sarcastic, as the number √(-1) was thought not to exist by many mathematicians. It was not until the work of Euler in analysis that the imaginary number i was properly understood and widely acknowledged as being a proper numberAnother AnswerMathematicians call the horizontal and vertical axes of a graph, the 'real' and 'imaginary' axes. Numbers lying along the real (horizontal) axis are called 'real numbers', and numbers lying along the imaginary (vertical) axis are called 'imaginary numbers'.(see first discussion page entry)
Complex math covers how to do operations on complex numbers. Complex numbers include real numbers, imaginary numbers, and the combination of real+imaginary numbers.