One such example is the notion of “negative numbers.” For centuries that expression was viewed as an oxymoron, a self-canceling phrase, a numerical absurdity; numbers count or measure things - there is no shape that has negative area, no circle with negative circumference, no book with a negative number of pages. For literally hundreds of years, great pains were taken to solve problems by methods that circumvented the use of negative numbers. When an answer to an equation came out to be negative, mathematicians either rearranged their equations or considered the answer to be “absurd” and tossed it out. Only very gradually did it become clear that the effort spent in avoiding their use was wasted effort, for negative numbers, though not interpretable in the same fashion as positive numbers, were just as acceptable, and in no way a contradictory system of numbers.
The notion of an imaginary number followed a similar pattern of initial rejection and gradual acceptance. The introduction of imaginary numbers was both an imaginative and risky act, since there was the possibility that the use of imaginary numbers could become more and more commonplace, and only much later lead to a serious contradiction in how systems of number are constructed, in which case all the previous work would have to be discarded. However, by the 19th century, when number systems were examined much more closely, it became clear that “imaginary numbers” were no more or less “real” than the standard “real numbers.” Both are mathematical abstractions, and the “real” numbers include not only negative numbers that had been looked upon for so long with great suspicion but also oddities such as infinite decimals that never repeat (irrational numbers) and that do not satisfy any algebraic equation (called transcendental numbers). And so imaginary numbers came to be fully accepted as part of a mathematician’s toolkit, available whenever needed to solve problems. Imaginary numbers are now routinely used by engineers and physicists, and many applications of mathematics would be unthinkable without them...
-The Poetry of the Universe, Robert Osserman
The year is 1545. The Italian mathematician Girolamo Cardano is trying to come up with a general solution to the cubic equation
as had been done for the general quadratic equation 
. He ends up with an answer that must include 
.......in 1545 square roots of negative numbers had no legitimacy and the theory of complex numbers was nonexistent. How should these meaningless symbols be interpreted? It was incompleteness and enigma. Mathematicians were curious. They needed to understand. The current mathematics had led them to a new problem. It would be almost three centuries before an adequate theory would be available to interpret properly and to legitimize this work.
The solution to the mystery was ultimately given in the early 1800s by accepting square roots of negative numbers and considering complex numbers to be points in a coordinate plane, whose horizontal axis is the real axis and whose vertical axis is the imaginary axis or i-axis.
Once we conceive of the real line as embedded in a plane of complex numbers, we have entered a whole new domain of mathematics. All our old knowledge of real algebra and analysis becomes enlarged and enriched when reinterpreted in the complex domain. In addition, we immediately see countless new problems and questions which could not even have been raised in the context of real numbers alone.
-The Mathematical Experience, Davis & Hersh