Pioneer of computing in Bulgaria and of abstract mathematics, Iliev was born in Samokov. He graduated from Sofia University in 1908, became a professor in 1925, and earned a PhD in Mathematical Sciences in Naples that same year. He was elected an academician in 1930 and was a member of mathematical societies in France, Czechoslovakia, Poland, Peru, and elsewhere.
His main contributions were in the field of functions theory, numeric theory, and quadratic equations. He examined integral representation and the application of Newton quotients, and he discovered and described the main characteristics and recurrent relations of their Peano nuclei. These findings became a basis of the theory of splayed functions, which took shape in the 1960s and is considered to be one of the notable discoveries of applied analysis in the second half of the twentieth century.
Iliev also found the exact range of the semi-interval where the undefined point of the Rolle theorem lies. The method applied by Iliev and further developed by Favard is known in the literature as the Chakalov–Favard method.
