Use this tag for questions concerning history of mathematics, historical primacies of results, and evolution of terminology, symbols, and notations.

learn more… | top users | synonyms (1)

575
votes
16answers
53k views

Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?

In the book Thomas's Calculus (11th edition) it is mentioned (Section 3.8 pg 225) that the derivative $\frac{\textrm{d}y}{\textrm{d}x}$ is not a ratio. Couldn't it be interpreted as a ratio, because ...
18
votes
1answer
1k views

About a paper of Zermelo

This about the famous article Zermelo, E., Beweis, daß jede Menge wohlgeordnet werden kann, Math. Ann. 59 (4), 514–516 (1904), available here. Edit: Springer link to the ...
75
votes
11answers
18k views

Is zero odd or even?

Some books say even numbers start from two but if you consider the number line concept, I think zero should be even because it is in between -1 and +1 (i.e in between 2 odd numbers). What is the real ...
136
votes
3answers
7k views

A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I

The approximation $$\sin(x) \simeq \frac{16 (\pi -x) x}{5 \pi ^2-4 (\pi -x) x}\qquad (0\leq x\leq\pi)$$ was proposed by Mahabhaskariya of Bhaskara I, a seventh-century Indian mathematician. I ...
37
votes
2answers
3k views

What concept does an open set axiomatise?

In the context of metric (and in general first-countable) topologies, it's reasonably clear what a closed set is: a set $F$ is closed if and only if every convergent sequence of points in $F$ ...
121
votes
1answer
11k views

Why are rings called rings?

I've done some search in Internet and other sources about this question. Why the name ring to this particular object? Just curiosity. Thanks.
15
votes
3answers
2k views

When the trig functions moved from the right triangle to the unit circle?

I have to write a paper about the unit circle and I'm trying to uncover some of its origins. Also, when the trig functions were expanded to angles greater than 90° and what was the rationale behind ...
172
votes
22answers
11k views

Why do mathematicians use single-letter variables?

I have much more experience programming than I do with advanced mathematics, so perhaps this is just a comfort thing with me, but I often get frustrated trying to follow mathematical notation. ...
9
votes
2answers
2k views

Notation for intervals

I have frequently encountered both $\langle a,b \rangle$ and $[a,b]$ as notation for closed intervals. I have mostly encountered $(a,b)$ for open intervals, but I have also seen $]a,b[$. I recall ...
84
votes
6answers
3k views

Why does mathematical convention deal so ineptly with multisets?

Many statements of mathematics are phrased most naturally in terms of multisets. For example: Every positive integer can be uniquely expressed as the product of a multiset of primes. But this ...
32
votes
3answers
5k views

Why, historically, do we multiply matrices as we do?

Multiplication of matrices — taking the dot product of the $i$th row of the first matrix and the $j$th column of the second to yield the $ij$th entry of the product — is not a very ...
55
votes
6answers
13k views

Why is a full turn of the circle 360°? Why not any other number?

I was just wondering why we have 90° degrees for a perpendicular angle. Why not 100° or any other number? What is the significance of 90° for the perpendicular or 360° for a circle? I didn't ever ...
8
votes
9answers
1k views

what is the definition of Mathematics ? [closed]

we all study mathematics , and all of us learn mathematical methods to solve problems , we learn how to prove , how to think mathematically but the question is, what is mathematics ? how can we ...
36
votes
2answers
3k views

Development of the Idea of the Determinant

While I basically understand what a determinant is, I wonder how this idea was developed? What was the principal idea behind its origination? I would like to know this so that I can have a better ...
11
votes
1answer
6k views

How were Hyperbolic functions derived/discovered?

Trig functions are simple ratios, but what does Cosh, Sinh and Tanh compute? How are they related to euler's number anyway?
40
votes
7answers
10k views

Good books on Math History

I'm trying to find good books on the history of mathematics, dating as far back as possible. There was a similar question here Good books on Philosophy of Mathematics, but mostly pertaining to ...
45
votes
3answers
3k views

History of the Concept of a Ring

I am vaguely familiar with the broad strokes of the development of group theory, first when ideas of geometric symmetries were studied in concrete settings without the abstract notion of a group ...
29
votes
4answers
6k views

Origin of the dot and cross product?

Most questions usually just relate to what these can be used for, that's fairly obvious to me since I've been programming 3D games/simulations for a while, but I've never really understood the inner ...
32
votes
1answer
580 views

Is the Galois group associated to a random polynomial solvable with probability 0?

Choose a random polynomial $P\in\mathbb{Z}[x]$ of degree $n$ and coefficients $\leq n$ and $\geq-n$. Let $r_1,\ldots,r_n$ be the roots of $P$ and consider ...
12
votes
2answers
2k views

Motivation for Napier's Logarithms

In the wikipedia article on logarithms, I am clueless about the approach and motivation for the following computations done by Napier (and the mysterious appearance of Euler's number) in this section. ...
17
votes
2answers
3k views

History of dot product and cosine

The fact that the dot product and the cosine of the angle between two vectors are mutually computable is easy to show (see the two sides in the two answers at Dot product in coordinates). But looking ...
179
votes
19answers
13k views

In the history of mathematics, has there ever been a mistake?

I was just wondering whether or not there have been mistakes in mathematics. Not a conjecture that ended up being false, but a theorem which had a proof that was accepted for a nontrivial amount of ...
63
votes
18answers
20k views

Anecdotes about famous mathematicians or physicists

I'm not sure whether this question suits this website, however, I don't know where else I could ask it. It is no mathematical problem or something similar, still I hope it won't be closed. A few ...
89
votes
16answers
10k views

Why did mathematicians take Russell's paradox seriously?

Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most ...
17
votes
10answers
1k views

Challenge: Demonstrate a Contradiction in Leibniz' differential notation

I want to know if the Leibniz differential notation actually leads to contradictions - I am starting to think it does not. And just to eliminate the most commonly showcased 'difficulty': For the ...
28
votes
1answer
10k views

Who discovered this number-guessing paradox?

In this math.se post I described in some detail a certain paradox, which I will summarize: $A$ writes two distinct numbers on slips of paper. $B$ selects one of the slips at random (equiprobably), ...
22
votes
2answers
2k views

Origins of the modern definition of topology

The modern definition of topology is 'a family of subsets of a set $X$ containing the empty set and $X$, closed under unions and finite intersections'. In Grundzüge der Mengenlehre (1914) Hausdorff ...
15
votes
2answers
4k views

“L'Hôpital's rule” vs. “L'Hospital's rule”?

I know this is not strictly a mathematical question, and I considered putting it on Linguistics SE, but I decided that seeing as this is most probably a mathematical history question, it would be ...
103
votes
34answers
17k views

Examples of mathematical results discovered “late”

What are examples of mathematical results that were discovered surprisingly late in history? Maybe the result is a straightforward corollary of an established theorem, or maybe it's just so simple ...
115
votes
10answers
8k views

Why do people use “it is easy to prove”?

Math is not generally what I am doing, but I have to read some literature and articles in dynamic systems and complexity theory. What I noticed is that authors tend to use (quite frequently) the ...
96
votes
8answers
14k views

Are half of all numbers odd?

Plato puts the following words in Socrates' mouth in the Phaedo dialogue: I mean, for instance, the number three, and there are many other examples. Take the case of three; do you not think it may ...
59
votes
6answers
4k views

What kind of “symmetry” is the symmetric group about?

There are two concepts which are very similar literally in abstract algebra: symmetric group and symmetry group. By definition, the symmetric group on a set is the group consisting of all bijections ...
38
votes
7answers
2k views

Why are topological spaces interesting to study?

In introductory real analysis, I dealt only with $\mathbb{R}^n$. Then I saw that limits can be defined in more abstract spaces than $\mathbb{R}^n$, namely the metric spaces. This abstraction seemed ...
42
votes
1answer
4k views

Was Grothendieck familiar with Stone's work on Boolean algebras?

In short, my question is: Was Grothendieck familiar with Stone's work on Boolean algebras? Background: In an answer to Pierre-Yves Gaillard's question Did Zariski really define the Zariski ...
15
votes
2answers
2k views

Why is it called Sylvester's Law of Inertia?

By "Sylvester's Law of Inertia," I mean: http://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia How does the name "Law of Inertia" fit with the statement of the theorem? I guess it's from ...
14
votes
8answers
2k views

Reference request: is mathematics discovered or created?

I have to write a short monograph as an assignment for a course on the philosophy of science. Being a math student, of course I want to opt for something math-related. After some initial ideas which ...
4
votes
3answers
1k views

Base 10 notation

Why are the ten digits in base 10 noted $0, 1, 2 ,3 ,4 ,5 ,6 ,7 ,8 ,9$ ? I heard it has to do with angles, but since I can't see how $9$ has $9$ angles, this may not be the historical reason ?
115
votes
33answers
16k views

Can you provide me historical examples of pure mathematics becoming “useful”?

I'm trying to think/know about something but I don't know if my basis premise is plausible, here we go. Sometimes when I'm talking with people about pure mathematics, they usually dismiss it because ...
74
votes
15answers
12k views

What is the smallest unknown natural number?

There are several unknown numbers in mathematics, such as optimal constants in some inequalities. Often it is enough to some estimates for these numbers from above and below, but finding the exact ...
53
votes
19answers
5k views

Theorems' names that don't credit the right people

The point of this question is to compile a list of theorems that don't give credit to right people in the sense that the name(s) of the mathematician(s) who first proved the theorem doesn't (do not) ...
45
votes
6answers
5k views

How hard is the proof of $\pi$ or $e$ being transcendental?

I understand that $\pi$ and $e$ are transcendental and that these are not simple facts. I mean, I have been told that these results are deep and difficult, and I am happy to believe them. I am curious ...
31
votes
1answer
1k views

History of “Show that $44\dots 88 \dots 9$ is a perfect square”

The problem Show that the sequence, $49, 4489, 444889, \dots$, gotten by inserting the digits $48$ in the middle of the previous number (all in base $10$), consists only of perfect squares. ...
21
votes
3answers
708 views

Approximation for $\pi$

I just stumbled upon $$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $$ which is $\delta = 0.0000481330$ different from $\pi$. Although this is a rather crude approximation I ...
19
votes
1answer
675 views

A curious theorem by Peano

Let $f$ be defined on $[a,b]$ and there differentiable. Show that for every $ \epsilon>0 $ there exists a partition $\, a=a_0<a_1<...<a_n=b \,$ of $ \,[a,b] \,$ so that $$\left|\frac ...
11
votes
6answers
9k views

Why do we need vectors and who invented it?

It is natural to understand the need for scalars (numbers), but why did we invent vectors? Who invented it and for what? EDIT: As George Lowther pointed out, the problem is too broad; I added the ...
18
votes
5answers
916 views

history of the double root solution of $ay''+by'+cy=0$

Motivation: It is a well-known fact that $ay''+by'+cy=0$ has solutions which are found from substituting the ansatz $y=e^{\lambda t}$ into the DEqn. It turns out that we replace the calculus problem ...
11
votes
5answers
1k views

What is the meaning of set-theoretic notation {}=0 and {{}}=1?

I'm told by very intelligent set-theorists that 0={} and 1={{}}. First and foremost I'm not saying that this is false, I'm just a pretty dumb and stupid fellow who can't handle this concept in his ...
12
votes
7answers
1k views

What's the hard part of zero?

A lot of textbooks said it was hard for human to accept zero when it was first introduced. How could it be? It seems to me as natural as positive integer which represent there is no elements at all.
9
votes
4answers
1k views

Uses of the 'Golden Ratio'

I have heard much about the numerous appearances of the ratio found in nature: 1.6180339887. Are there any actual mathematical uses that have been found of this number? What are its advantages? Just ...
7
votes
3answers
537 views

How was $e$ first calculated?

I understand how $\pi$ is calculated, but I am interested in references that explain when and how the natural exponent $e$ was developed. What mathematical principles are behind the value of $e$?