Were complex numbers ever considered to be of limited usefulness that is not very useful in practice (unlike modern science where we strongly need complex numbers)?

Note my question is not about philosophical consideration of complex numbers as something "imaginary" (existing only in imagination of people).

  • $\begingroup$ "First" complex numbers were not considered at all, until 16th century. Cardano does a single example with them, a formal manipulation. They slowly entered algebra and analysis in 17-18th centuries exactly because mathematicians found more and more uses for them (for solving algebraic equations, or studying analytic functions, for example). $\endgroup$ – Conifold Apr 18 '17 at 20:30

Usefulness was never seriously questioned. Complex numbers are useful, for example, if you want to make Cardano's formula work. And this is how they were invented. They are useful for other mathematical problems, and since the beginning of 19th century, in physics too.

What was questioned sometimes is their justification, existence. These questions gradually disappear when their geometric interpretation became a common knowledge, which happened in the beginning of 19th century. But still some mathematicians of 19th century (Chebyshev, for example) distrusted them and tried to avoid them.

I read somewhere (don't remember where) that students of Ecole Polytechnique once in the beginning of 19th century made a riot protesting teaching complex numbers (on the grounds that they are useless in practical applications), if this story is true, these were just poorly educated students. Serious mathematicians never doubted that they are useful.


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