# Tag Info

Accepted

### Has the idea that the result of division of positive number by negative number should be negative ever been controversial?

Much has been written about various roadblocks to the acceptance of negative numbers, and I have a folder containing photocopies of a few such papers, but I don't have time now to look for that folder....
• 3,532

### Who discovered the indeterminate forms like 0/0?

Special cases were handled algebraically even before the "l'Hopital's" rule, which appears in l'Hopital's 1696 transcription of tips on calculus he purchased (literally) from Johann Bernoulli in 1694, ...
• 77.6k
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### How was the sum of squares formula discovered by Archimedes?

Although it is commonly said that Archimedes summed the geometric series in Quadrature of the Parabola, and summed squares of integers in On Conoids and Spheroids and On Spirals, that is not what he ...
• 77.6k
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### Etymology of Some Terms Used in Ratio and Proportion in Old Algebra Textbooks

These terms derive from the four Latin verbs invertere "to turn upside down", alternare "to alternate, to arrange in alternating order", componere "to add together", and dividere "to divide up, to ...
• 6,704

### Why was the cubic specifically so hard to solve?

There is nothing special about cubic. I think the right question is: why the quadratic equations are much easier to solve than cubic and quartic? Well it's pretty obvious, is not it? Quadratic is ...
Accepted

### When were polynomial equations first factored?

I found no trace of factoring polynomials being done before Descartes, in his La Géométrie, written in 1637. In it, he wrote: It is evident from the above that the sum of an equation having several ...
• 5,907

### Who invented multiplying by the conjugate to rationalize denominators and when?

As with many "who was first" questions there is no straightforward answer. This is what May described as priority chasing coming to grief in Historiographic vices: "The hope of finding ...
• 77.6k

### Why was the cubic specifically so hard to solve?

The primary reason is that you have a roadblock: you absolutely need complex numbers - even for real-valued solutions - unless you go the route of using trigonometry and/or hyperbolic functions. The ...
• 229
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### Were ancient Romans so bad at computations before Arab numerals?

No, Romans were not at all bad at computations before Arab numerals were introduced to them. In fact, Romans had a perfectly fine way of doing computations that was every bit as good as Arabic ...
• 2,187

### What mathematical developments/discoveries caused imaginary numbers to gain acceptance at the time (18th century) they did?

Apart from the necessity in the calculation of roots of cubic polynomials, there is another, more fundamental role complex numbers play in polynomial equations, which was only beginning to be ...
• 709
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### Who was the first to show that this quintic equation has five radical roots?

I originally answered this on a different forum*. BK's solution (with some minor improvements by GW) is: $$x=\sum_{i\geq 0}\frac{(-1)^{i}}{i! (i m+1)}\prod_{j=0}^{i-1}(\frac{i m+1}{n}-j).$$ It ...
• 156

### How was the sum of squares formula discovered by Archimedes?

You are right about the person (Archimedes), but not about the work. He stated it as proposition 10 of his text On spirals. In modern language, the statement is(\forall n\in\mathbb{N}):(n+1)n^2+(1+2+...
• 5,907
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### Where can I find the list of problems from the (Chinese) "Nine Chapters on Mathematical Art"?

From what I understand the "list of problems" is supposed to be the full English translation. Here is the bibliographic reference for the 1999 translation, the only one so far: The Nine ...
• 77.6k

### Do these trigonometric identities belong to Antonio Cagnoli?

Your first identity is indeed attributed to Cagnoli by e.g. Franchini (1805), Encyclopédie du dix-neuvième siècle (1847), or Le Cointe (1858, p. 59). It is in the second edition of his Trigonometria (...
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### What is the origin of the method of undetermined coefficients?

The method of undetermined coefficients was probably first developed by Euler, he applied it in his work to analyze the perturbations of Saturn by Jupiter for the Paris Academy's prize competition of ...
• 4,600
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### Where did Lagrange write his technique of resolvents for solving polynomials?

The relevant publication seems to have been published across two annual volumes (for the years 1770 and 1771, but likely printed in 1773) of the Proceedings of the Royal Academy of Science and ...
• 6,704

### Why was solving polynomial equations historically considered so interesting?

The conics as geometric shapes were investigated comprehensively by many ancient Greek mathematicians. After Descartes's innovation of introducing coordinates, they were seen as essential examples of ...
• 3,758
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### Why did al-Khwarizmi use al jebr, ‘the reuniting of broken parts’ to signify algebra?

See Roshdi Rashed, Classical Mathematics from Al-Khwarizmi to Descartes (Routledge, 2014), page 107 [but I've not seen: R. Rashed (editor), Al-Khwārizmī: The Beginnings of Algebra (2009)]: The term ...
• 14.9k

### Why did al-Khwarizmi use al jebr, ‘the reuniting of broken parts’ to signify algebra?

I have also wondered about the term al-jabr which ultimately became algebra. Re Please see Solomon Gandz (1926) The Origin of the Term “Algebra”, The American Mathematical Monthly, 33:9, 437-440, DOI: ...
• 4,079
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### How did Ruffini discover his method of polynomial division?

Polynomial division algorithms were known long before Ruffini's Sopra la determinazione delle radici (1804). For a history of even older numerical division algorithms see Who invented short and long ...
• 77.6k
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### Who first introduced the longhand square-rooting method into European mathematics?

Whenever you wonder about who first introduced something mathematical into Western Europe, Leonardo of Pisa aka Fibonacci should be the first person you think of. Fibonacci's 1202 work Liber Abaci has ...
• 602

### When was the idea of exponents generalized from "repeated multiplication"?

This was essentially done by Napier, see Which came first, the natural logarithm or the base of the natural logarithm? and How did Napier come to invent logarithms? Napier was talking about the ...
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### Why does the "Principle Of Permanence" have two different definitions?

I've come to the conclusion that the Wikipedia article is simply wrong, and I've completely rewritten it (permalink to original). It described the Identity Theorem, not any kind of "Principle of ...
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### Why does the "Principle Of Permanence" have two different definitions?

Yes, there are at least four somewhat-different things that have either been called a "principle of permanence" historically, or might be called so currently. The easiest to explain (and ...
• 1,096

### Why did al-Khwarizmi use al jebr, ‘the reuniting of broken parts’ to signify algebra?

I published an article on the meaning of the terms al-jabr and al-muqabala in Arabic algebra, which you can download from my Academia.edu page: "Simplifying equations in Arabic algebra" ...
1 vote

### Why was solving polynomial equations historically considered so interesting?

The study of the quintic by mathematicians such as Lagrange, Abel and Galois led to profound insights that extend well beyond polynomial equations. For example, Galois' proof of the non-existence of ...
1 vote

### Who was the first to show that this quintic equation has five radical roots?

It's not at all clear what this question is asking. The fact that the author posted the question 5-6 years ago, and only just replied to a comment, is also very odd. It probably doesn't deserve much ...
• 1,446
1 vote
Accepted

### Why $x_a$ (or $x_o$) and not $a_x$? (conventions for algebraic quantities)

Like Ben Crowell in the comments, I'm not sure I fully understand the question. But interpreting it broadly about the origin of the index notation, I quote Spalt who in Die Analysis im Wandel und im ...
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1 vote

### Why $x_a$ (or $x_o$) and not $a_x$? (conventions for algebraic quantities)

If we write $0_x$ then it looks like a special kind of 0, since the 0 is above the baseline for typography. So we would expect $0_x+1_x=1_x$ etc. With the $x_0$ notation it is clear that $x_0$ is a ...

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