# Tag Info

19

The first serious use of complex numbers is in finding the roots of quadratic, cubic, and quartic polynomials. Cardano, in his Ars Magna (1545), first showed that quadratic equations could have (formally) complex roots, although he didn't call them that; he said they were "as subtle as [they are] useless". In Bombelli's algebra text (1572), he developed the ...

18

There is indeed a difference between 18th century and modern concepts of algebra. For example, Lagrange and his contemporaries did not define algebraic structures into existence by specifying axioms for operations on abstract sets. Most of algebra was about integers and polynomials, which were assumed to pre-exist, and they were not thought as manifestations ...

10

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, see Indeterminate Forms Revisited, by Boas. For example, Descartes's method of finding tangents involved resolving "indeterminate forms" like $0/0$, see Is ...

9

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 did. He "sums" not even areas of figures (as in numbers associated to them) but figures themselves, and he certainly does not relate any of that to stacking ...

8

The answer depends on what you consider the "modern concepts". In modern algebra, they usually do not consider polynomials or rational functions as functions of the variable $x$, but exactly as you say in $1$, as elements of the field $C(y)$. So Lagrange's writing is not more symbolic/abstract than modern algebra, but more abstract than modern "high school ...

7

I will try to answer this both in 18th century terms, and in modern dress. First off, the réduite equation was introduced on p.208 and named on p.213; it is $$y^6 + py^3 - n^3/27 = 0$$ The coefficients of this equation are $p$ and $-n^3/27$, thus depending only on the coefficients $n,p$ of the original equation $x^3+nx+p=0$. (Lagrange briefly allows an $... 7 These terms derive from the four Latin verbs invertere "to turn upside down", alternare "to alternate, to arrange in alternating order", componere "to add together", and dividere "to divide up, to separate into parts". The specific grammatical form is the gerund, indicated by -nd-, and the grammatical case used is the ablative, indicated by the suffix -o. ... 6 I found no trace of factoring polynomials being done before Descartes, in his La Géométrie, written in 1637. In it, he wrote: It is evident from the above that the sum of an equation having several roots [that is, the polynomial itself] is always divisible by a binomial consisting of the unknown quantity diminished by the false roots. In this way, the ... 6 Just to complement Danu's answer. Some people used complex numbers since the 16th century, however, WIDE acceptance came later (at the end of the 18th century) when several people (Argand, Vessel, Gauss) discovered the geometric interpretation. This was apparently a crucial step. Still, they were not universally recognized. They say that even Chebyshev ... 6 An early example is Arithmetick Made fo Easy, That it may be Learned without a Master: After a new and concife method; the Like not yet Extant The linked second edition was published in 1740. The first edition was 1727. (1725 for the original French: L'arithmétique rendue facile de façon à la pouvoir apprendre sans Maître) The translator's preface ... 5 Apart from the necessity in the calculation of roots of cubic polynomials, there is another, more fundamental role complex numbers play in polynomial equations, which was only beginning to be appreciated in the 17th century. This role is expressed through the fundamental theorem of algebra, which says that any nonconstant polynomial equation has at least ... 5 You can find it in Ramanujan's Notebooks IV by B. Berndt, Chap. 22 Elementary Results, Entry 20, p.31. Ramanujan starts with this problem. Let$a,b,c,d$be arbitrary. Solve the system, $$x^2+ay = b\tag{20a}$$ $$y^2+cx = d\tag{20b}$$ Eliminating$y$, we find it is equivalent to, $$a^2(d-cx) = (b-x^2)^2\tag{20.1}$$ Assume without loss of generality that$...

5

For a broader perspective see How was geometry historically used to solve polynomial equations? For early practical problems that would lead (today) to quadratic equations see e.g. Friberg's discussion of cuneiform tablet YBC 3879 (c. 2000 BC), a judicial document from third Sumerian Ur period, that describes field division problems leading to quadratic ...

4

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 a 'first' comes to grief because of the historically dynamic character of ideas... If we describe a result with sufficient vagueness, there seems to be an ...

4

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$$(\forall n\in\mathbb{N}):(n+1)n^2+(1+2+\cdots+n)=3(1^2+2^2+\cdots+n^2).$$See page 162 of this edition of the works of Archimedes.

4

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 Chapters on the Mathematical Art: Companion and Commentary. By Shen Kangshen, John N. Crossley, and Anthony W.-C. Lun. Oxford and New York: Oxford University Press, ...

4

Up to notational conventions polynomial division algorithm was first described by an Arabic mathematician al-Samawal (1130-1180), who can also be credited with defining what we call polynomials today, see Who invented short and long division? In light of that the factorization theorem does not predate polynomial division.

3

As you correctly noted, the first "principle or law of the permanence of equivalent forms" was formulated by the English algebraist George Peacock in his book A treatise of Algebra (1st ed.1830), page 104 : "Whatever form is Algebraically equivalent to another, when expressed in general symbols, must be true, whatever those symbols denote." Peacock was ...

3

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 inverse function, but the tables he made can (and have to) be used both ways. The formal definition of the exponent for all numbers, even for complex ones is due to ...

3

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 division? Some authors, see also Victor Katz's History of Mathematics (7.2.3), credit medieval Arabic mathematician al-Samaw'al (1130-1180) for inventing long ...

3

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 (1804, Chap. IV, nº 173) but apparently not in the first (1786) where Chap. IV ends at nº 128. Woodhouse (1819) writes: $$\tan.(A+B+C)=\frac{t+t'+t''-tt't''}{... 3 Whenever you wonder about who first introduced something mathematical into Western Europe, Leonardo of Pisa aka Fibonacci should be the first person you think of. Fibonacci's 1202 work Liber Abaci has a chapter dedicated to the finding of square and cube roots, with an introduction referencing Euclid and Al-Khwarizmi, and then launching into a demonstration ... 3 I originally answered this on a different forum*. BK's solution (with some minor improvements by GW) is:$$x=\sum_{i\geq 0}\frac{(-1)^{i}}{i! (i m+1)}\prod_{j=0}^{i-1}(\frac{i m+1}{n}-j). It appears to be the first discovered general formula for real roots of the polynomial $x^n + x^m - 1 = 0$. It is not clear whether other radical roots can also be ...

1

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 numerals. They used a table and stones to do computations, similar to an abacus. Roman numerals were just used to record numbers, not to do long computations. Though ...

1

I I think Fermat had the proof for n=3 based on the method of infinite descent.Pl.read on the internet'method of infinite descent and proof of Fermat's last theorem for n=3" This proof is in Netherlands text book on FLT in general.

1

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 special kind of $x$, or at any rate the same kind of thing as $x$.

1

If you read the papers under with keywords "Holomorphic dynamics", "Mandelbrot set", and MLC conjecture, you discover that quadratic functions are still a hot research topic. A. Douady and J. Hubbard recalled in the late 90s, that when in the early 80-s they were asked "what problems are you working on now ?", they replied "we study quadratic polynomials (of ...

1

The principle of permanence was a heuristic principle serving an important role for discovery of new results, historically speaking. In modern mathematics, it has been formulated as a rigorous logical principle called the transfer principle. This exists in a number of contexts but is most famously present in Robinson's framework which involves a hyperreal ...

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