# When was the square of negative numbers specified?

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?

• But the square of $(-a)$ is $(-a) \times (-a)$. – Mauro ALLEGRANZA Apr 15 '18 at 18:32
• @user157860 This is a fundamental question that must be investigated fully in depth, otherwise it seems like a plain belief that would fall somewhere else, good question – Bassam Karzeddin Apr 16 '18 at 7:18
• Are you asking when someone realized that, if (-1)* X yields a negative number, then (-1)*(negative number) must be positive? – Carl Witthoft Apr 16 '18 at 12:00
• Well, user157860, that's a radically (pun intended) different question from the one you posted. – Carl Witthoft Apr 16 '18 at 15:36
• Possible duplicate of Historically, how did people define multiplication for negative numbers? – Conifold Apr 16 '18 at 20:51

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...

• The problem of $\sqrt{-1}$ is a different problem. It occured when those Italians discovered the cubic formula and found that this formula contains square roots of negative numbers, even when the final result is real. – Alexandre Eremenko Apr 16 '18 at 18:22
• Yes, I have access to his books and he says this in it explicitly. – Alexandre Eremenko Apr 19 '18 at 11:57
• I added my literal translation from Diophantus. On your second question, I just don't understand it. The rules of multiplication of complex numbers are no more "ad hoc" then the rules of multiplication of real numbers. And complex numbers "exist" in the so-called "real world" in the same sense as real numbers, or rational numbers for that matter. But I am not going to discuss philosophy on this site. – Alexandre Eremenko Apr 19 '18 at 18:15
• @user157860: $a^2=a\times a$, so a square is a special case of multiplication, is not it? – Alexandre Eremenko Apr 20 '18 at 12:23
• It surely is for natural numbers, bu we are concerned about numeri (ab) "surdi" here – user157860 Apr 21 '18 at 5:22

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.

• Cardano, Bombelli, and others were certainly aware of the problem in the geometrical sense. The side of a square was always positive, so the square's area was always positive - hence it made no sense to start with a negative area (and then determine that a square had a negative side by taking the square root). Unfortunately I don't have an English translation of Bombelli's L'Algebra, but I know he discussed negative numbers to some extent. I'll scan the various other history books I have and see if any of them provide any enlightening detail. I'm curious too. – Brant Apr 21 '18 at 11:50
• I meant in a geometrical sense and in Cardano's time. What is the meaning of a silo capable of storing -125 bushels of grain? No such silo exists in the physical world. Things are admittedly different in the algebraic world. Cubes in real life - physical, structural cubes - don't have negative volumes or negative side lengths. Yet the rules for multiplying negative numbers dictate that that -5 x -5 x -5 = -125. So, algebraically, the cube root of a negative number has a real solution, but the square root of a negative number does not. – Brant Apr 22 '18 at 18:58
• Historical roots of the justification for the rule for multiplication of negative numbers has some good info that is related to this discussion. – Brant Apr 22 '18 at 21:34