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I am going over some history of the complex numbers, and two things baffle me (and they are not mathematics).

  1. From Cardano's time to around the 18th century, negative numbers were not accepted by all mathematicians, and there was absolutely no universal or formal/proven rules of arithmetic with them, maybe it was all just “word of mouth” or “experimentation”?
  2. How can you study the square root of a negative if you don't even accept negatives?
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    $\begingroup$ By 17th century, there were almost universally accepted formal rules for negatives, regardless of whether they were "accepted" as numbers or not, and Wallis's number line visualization. By 18th century, they were codified in textbooks, Maclaurin's, Euler's, etc. Imaginaries were treated in a formal way only until Argand's plane, with gloves and disputes over extending other operations to them, like exponentiation and logarithms. Peacock later termed this formal attitude of expanding algebraic rules the "principle of permanence". $\endgroup$
    – Conifold
    Commented Feb 19 at 7:06

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Those mathematicians who worked with complex numbers, of course did accept negative numbers. When you say that "negative numbers were not accepted", you mean that they were not UNIVERSALLY accepted. Neither complex numbers were. At any stage of development some people are much more advanced than other people.

If you ever filed a US tax return, you know that IRS does not accept negative numbers. Otherwise how do you explain their instructions: "If line X is less than or equal to line Y, subtract line X from line Y and enter here". And later "If X is greater than Y, subtract Y from X and enter here".

Some historian 500 years from now may conclude that in 21st century negative numbers were not accepted in the US.

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