The statement presented above, in Russian algebra school textbooks is named Vieta's theorem. I was not succeeded to find this assertion in Vieta's work. I only found the statement that, using modern notation, the numbers a and b are the roots of the equation x^2 - (a+b)x + ab = 0; what, according to the mentioned books, is the inverse Vieta's theorem.
I ask this question because, as I have established in my last works (presented here, in RG), the proposition which is ordinarily named the Vieta's theorem, is erroneous. So, it turns out that in the mentioned books, the Vieta's name is associated with non-true assertion. That is why it is highly desirable to identify the person who was the first onto this way.
The relationship between the roots of a quadratic equation and its coefficients has been known since ancient times, but the first recorded use of the specific formula you mentioned can be attributed to the Persian mathematician, al-Khwarizmi (c. 780–c. 850). In his book "Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala" (The Compendious Book on Calculation by Completion and Balancing), which was later translated into Latin as "Algoritmi de numero Indorum" and became the basis for algebraic notation, al-Khwarizmi discussed various methods for solving quadratic equations and stated the formula "If two quantities are equal to two other quantities, their sum is equal to the sum of those other two quantities. Let x and y be the roots of the equation x^2 + bx = c. Then, since x + y = -b, it follows that xy = c."
Regards,
Dear Ajit Singh,
Thank you very much for your answer. However, did you read the second paragraph of the text following for my question? I write there that the assertion mentioned in my question is wrong, although it is ordinary named the Vieta's theorem. Well, turn to the question,
Is it true that if x_1 and x_2 are the roots of a square trinomial x²+px+q, then x_1+x_2=-p and x_1x_2=q?
Everyone will answer “yes” to the above question. And will be mistaken (although, it is hard to find a school algebra textbook not containing this assertion under the name of Vieta’s theorem). To make this sure, take some specific trinomial, say, x²-5x+6. Direct computations show that, for all x, x²-5x+6=(x-2)(x-3). This means that the numbers 2 and 3 are the roots of the given trinomial, and no other numbers are its roots. Now,
Let x_1 and x_2 be the roots of the given trinomial.
Is it true that (x_1=2 and x_2=3) or (x_1=3 and x_2=2)?
No!
To see this, let us verify honestly, when it is true that x_1 and x_2 are the roots of the given trinomial. Therefor, let
x_1 and x_2 be the roots of the given trinomial. Then
(x_1–2)(x_1–3)=0 and (x_2–2)(x_3–3)=0, i. e.
(x_1=2 or x_1=3) and (x_2=2 or x_2=3), i. e.
a) (x_1=2 and x_2=2) or b) (x_1=2 and x_2=3) or c) (x_1=3 and x_2=2) or d) (x_1=3 and x_2=3).
Thus, the premise of the implication in the title of my question, for the given trinomial, is true precisely in four cases, namely, a), b), c), and d).
However, everyone can easily verify that the conclusion of the implication mentioned holds only in two of these four cases, namely, b) and c), while in the other two cases it is wrong. So, in the whole, the implication considered is wrong, as it was stated.
One can ask, whether this means that Vieta was mistaken? No. The matter is that Vieta did not prove and even formulated the statement, which is ordinarily called by his name. In his The Analytic Art, Vieta formulated without proof (as something obvious) the assertion, which is now called the inverse Vieta’s theorem.
Now, my question consists of, who was the first who formulated this wrong statement and called it Vieta's theorem?
At last, let us turn to your quotation from al-KHwarizmi. How to understand his words "two quantities are equal to two other quantities"? Let the first two quantities be x and y, the second two be u and v. How to understand that quantities x and y are equal to quantities u and v?
Regards,
A. Naziev
It is difficult to pinpoint the exact origin of the mistaken statement that x_1+x_2=-p and x_1x_2=q for a square trinomial x²+px+q, as it has been taught as Vieta's theorem in many algebra textbooks for centuries. However, it is important to note that Vieta himself did not formulate or prove this statement, and instead formulated a related theorem known as the inverse Vieta's theorem.
As for al-Khwarizmi's statement "two quantities are equal to two other quantities," it likely refers to a general equation of the form ax + b = cx + d, where x represents some unknown quantity. Al-Khwarizmi's statement implies that the two quantities on one side of the equation are equal to the two quantities on the other side, which can be used to solve for the unknown quantity x.
Regards,
Dear @Ajit Singh,
Thank you very much for your reply. It is very pleasant for me to hear that you agree with my thoughts I expressed in my previous letter, namely,
1) that the assertion ordinarily called Vieta's theorem is wrong;
2) that Vieta himself did not formulate or prove this statement, and
3) that Vieta in fact formulated the theorem now known as the inverse Vieta theorem.
However, it is hardly for me to agree with your interpretation of al-Khwarizmi statement as related to the systems of two linear equations with two unknowns, and almost impossible (at least, at present) to agree with the opinion that it is related to the square equations.
Meanwile, I have already published my mentioned thoughts in the Medium, and in the Academia.edu.
Regards,
Prof. A. Naziev
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