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I was reading “A brief history of numbers” by Corry, but I came across a part that confused me.

Cardano accepted the law of signs for “subtractions” proposed by an older group of Italian mathematicians because of the product of two binomials.

The author writes that that the quadratic equation $x^2 = px + q$ and two formulas are an example that force Cardano into accepting negatives but I don't see how this is the case.

If anyone else has any other examples or commentary it would be appreciated.

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  • $\begingroup$ I don't have my copy of Corry's book with me at the moment, but it would be helpful if you could let me know on which page Corry argues this point. My recollection is that although there was no available geometric interpretation of negative numbers, Cardano was able to verify these "false" (negative) solutions numerically and therefore was in a sense "forced" to accept their validity. Historically, negative numbers were often validated in relation to debits and credits. $\endgroup$
    – nwr
    Feb 14 at 15:23
  • $\begingroup$ For Cardano and Bombelli, see also Roy Wagner, The geometry of the unknown Bombelli's algebra linearia (2010) $\endgroup$ Feb 14 at 15:38
  • $\begingroup$ For a good discussion, see David Mumford, What is so Baffling about Negative numbers (2010) $\endgroup$ Feb 14 at 15:57
  • $\begingroup$ @nwr it appears Page 144 $\endgroup$
    – Fraser
    Feb 14 at 21:29

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