When was the square of negative numbers specified?

Question

We know that the rules of relative number where laid down in India (a product of 2 debts is a fortune) and in Europe they were spread by Bombelli, who , again, only mentions the product of two minuses.

When/by whom was it first specified that the square of a negative is a positive?

Edit

the question ha been misinterpreted and id is not a duplicate: someone is taking for granted that the rule of minus times minus authomatically implies the realization that negative squares are impossible (besides the subtle fact that another conclusion was theoretically possible).

So , Bombelli mentioned the rules of multiplication, but did he explicitly state that the square of minus one (or any other negative) is plus one? When was the general public of scientists fully aware that the roots of negatives are missing on the number line on the left of zero?

Surely not in the Middle ages before Bombelli even though (as the answer here and there imply) negatives where known from the 6th or 3rd century or even earlier. Is this clear now?

Answer 1

The oldest surviving book on algebra is "Arithmetic" by Diophantus of Alexandria. He defines negative numbers and arithmetic operations on them, including multiplication. So he knew that the product of negative numbers is positive.

Earlier sources on algebra did not survive, but there is little doubt that they existed: it is hard to imagine that such advanced mathematics appeared suddenly out of blue. Unfortunately, it is not known precisely when Diophantus lived. The accepted date is around 250 AD, but certainly he could not write after Theon (mid 300s) because Theon mentions his book.

Now, in the comment you ask a different question: when and why did people start to discuss square ROOTS of negative numbers. This happened in Italy, in 16th century when they discovered a formula for solving the cubic equation. When you apply this formula to a cubic equation which has 3 real roots, some intermediate term which you obtain is a square root of a negative number. So the question was how to interpret this root and to make the formula work. For details, see http://www.math.purdue.edu/~eremenko/dvi/cardano.pdf

EDIT. I was asked for the exact citation of Diophantus. The copy that I have is in Russian, so I translate from the Russian to English as literally as I can:

Arithmetic, book I, section IX (p. 40 of my Russian edition):

Deficiency multiplied on deficiency gives an asset; deficiency multiplied on an asset gives a deficiency; we use the following sign for a deficiency

sorry, I have no font to reproduce his sign for the minus. The words which I translated as "deficiency" and "asset" can be also translated as "lack" and "availability". Section X:

After this explanation of multiplication the division must be clear; it is recommended to the beginner to exercise in addition, subtraction and multiplication of these kinds...

Answer 2

In his 1784 work Algebra, Colin MacLaurin presents the following argument for why a negative number multiplied by a negative number is (or rather, must be) positive (see Chapter III case IV, here; it's page 35 of the PDF).

$-n(a-a)$ must equal 0 (since $a-a = 0$)

Using the distributive property, the first term $-n \times a$ is equal to $-na$. The only way for the distributive property to still hold, and the statement to be true, is if $-n \times -a=na$.

If we let $-n=-a$ from our example above, then $-a \times -a$ will of course be a positive number, $a^2$.

From this it is clear that there is no way to square a real number and end up with a negative result, hence Cardano's (and everyone else's) confusion over what to do when confronted with something like $\sqrt{-n}$: such an operation was undefined, because there wasn't any number at the time that could be squared to get a negative result.

In Book I of L'Algebra (1572), Bombelli specifies that "minus times minus makes plus", and even offers an example that is farily close to MacLaurin's:

Multiply $(6-4) \times (5-2)$

$-2 \times -4 = 8$, and

$-2 \times 6 = -12$, and

$5 \times -4 = -20$, and

$5 \times 6 = 30$

so $(6-4) \times (5-2) = 30-20-12+8$

Bombelli does not take the extra step to explain that a negative times a negative must be positive for the calculation to work out properly, however from this example and his multiplication rules he would have realized that (a) the square of a negative number is positive, and (b) thus there was no way to square a number and get a negative result, rendering the square roots of negative numbers perplexing at best. (A full version of L'Algebra in Italian can be found here. The above excerpts are from Book I {Libro Primo}, pages 70 and 71 {127 and 128 of the PDF}. Without knowing you at all, I bet your Italian is better than mine...)

I will say that I am not 100% certain that MacLaurin was the first one to actually demonstrate this "minus times minus is plus" rule (versus just stating it). Bombelli gave an example, but MacLarin's Treatise is the earliest publication I have found that offers something like a proof. I whole-heartedly invite the pros here to fact check me. I do hope that I have addressed the general spirit of your question, though.