Use the convention tag for questions about standard, cultural practices in mathematics.

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24
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9answers
2k views

What could be better than base 10?

Most people use base 10; it's obviously the common notation in the modern world. However, if we could change what became the common notation, would there be a better choice? I'm aware that it very ...
16
votes
6answers
472 views

Is there a fundamental reason that $\int_b^a = -\int_a^b$

Is there a fundamental reason that switching the order of the limits in an integral results in the negative, i.e., $$\int_b^af(x)\,dx = -\int_a^bf(x)\,dx?$$ As far as I can tell, this is just chosen ...
15
votes
3answers
532 views

Starting sentences with mathematical symbols.

I apologise if this is a duplicate in any way or is too opinion-based. To what extent is it best not to start a sentence with a mathematical symbol? I find that when trying to solve a problem or ...
13
votes
6answers
2k views

No radical in the denominator — why?

Why do all school algebra texts define simplest form for expressions with radicals to not allow a radical in the denominator. For the classic example, $1/\sqrt{3}$ needs to be "simplified" to ...
10
votes
3answers
465 views

Advocating base 12 number system

I had a calculus professor who suggested we should be using base 12 number system. What are the advantages of using such a system?
9
votes
5answers
501 views

Set {1,1} = Set {1}, origin of this convention

Is there any book that explicitly contain the convention that a representation of the set that contain repeated element is the same as the one without repeated elements? Like $\{1,1,2,3\} = ...
9
votes
5answers
439 views

What are reasons why some symbols in mathematical logic are not standardized?

Why is so hard to find a standardisation regarding symbolism and/or terminology in Mathematical Logic ? We see again and again students asking if e.g. $\rightarrow$ and $\implies$ means the same ...
9
votes
4answers
181 views

Why are the order-of-operations conventions good?

Children are sometimes taught silly mnemonics like "PEMDAS" to remember conventions on order of operations. (I never heard of "PEMDAS" until long after graduating from college, as far as I can ...
8
votes
5answers
791 views

use of $\sum $ for uncountable indexing set

I was wondering whether it makes sense to use the $\sum $ notation for uncountable indexing sets. For example it seems to me it would not make sense to say $$ \sum_{a \in A} a \quad \text{where A is ...
8
votes
3answers
219 views

Rationale for a convention: Why use the semiperimeter in Heron's formula?

Heron's formula says that the area of a triangle whose sides have lengths $a, b, c$ is $\sqrt{s(s-a)(s-b)(s-c)}$ where $s=(a+b+c)/2$ is the semiperimeter. It can also be stated by saying that the ...
8
votes
1answer
111 views

Wikipedia claims that number theorists tend to prefer $0 \notin \mathbb{N}$. Is this actually true?

According to wikipedia here: $\mathbb{N}$ means either $\{ 0, 1, 2, 3, ...\}$ or $\{ 1, 2, 3, ...\}.$ The choice depends on the area of mathematics being studied; e.g. number theorists ...
7
votes
2answers
343 views

Why does this text insist on changing the variable name here?

In What is mathematics? by Courant, Robbins, and Stewart, "5. An important inequality", the authors change $n$ in this example: $$(1+p)^n\geq1+np$$ to $r$ in this example: $$(1+p)^r\geq1+rp$$ In ...
6
votes
3answers
1k views

What does a “convention” mean in mathematics?

We all know that $0!=1$, the degree of the zero polynomial equals $-\infty$, the interval$[a,a)=(a,a]=(a,a)=\emptyset$ ... and so on, are conventions in mathematics. So is a convention something that ...
6
votes
2answers
180 views

Good Hygiene in using Quantifiers

When using quantifiers it is probably important to pick up certain habits that Veterans agree upon as early as possible. Since it was pointed out to me by a highly esteemed member that it's ...
6
votes
4answers
147 views

Zero polynomial [duplicate]

Possible Duplicate: Polynomial of degree $-\infty$? Today in Abstract Algebra my instructor briefly mentioned that sometimes the zero polynomial is defined to have degree $-\infty$. What ...
5
votes
1answer
139 views

The Degree of Zero Polynomial

I wonder why the degree of the zero polynomial is $-\infty$ ? I heard that, it is $-\infty$ to make the formula $\deg(fg)=\deg(f)+\deg(g)$ hold when one of these polynomials is zero. However, if that ...
4
votes
2answers
81 views

A question on notation: $u(x,y)$ vs. $u(x+iy)$.

In Tao's complex analysis notes, he uses the convention that $$f(z)=u(z)+iv(z)= u(x+iy)+iv(x+iy)$$ implying that the functions are $$u,v:\mathbb{C} \to \mathbb{R}.$$ But in other places I see the ...
4
votes
1answer
115 views

Where should the 2$\pi$ go in the Fourier Transform?

In some lecture notes on Harmonic Analysis from Terence Tao here, he defines the fourier transform by $$\hat{f}(\tau)=\int_{\mathbb{R}}e^{-2\pi i t\tau}f(t)dt$$ and then says This is really the ...
4
votes
3answers
91 views

Notation for $X - \mathbb{E}(X)$?

Let $X$ be a random variable with expectation value $\mathbb{E}(X)=\mu$. Is there a (reasonably standard) notation to denote the "centered" random variable $X - \mu$? And, while I'm at it, if $X_i$ ...
4
votes
3answers
442 views

What is a Neighborhood?

Which of these definitions is more commonly used, and in which contexts? Fix a point $x\in (X, \tau)$. Then a neighborhood around a point $x$ is: a set $N\ni x$ and $N\in \tau$ a set $N$ with $x\in ...
3
votes
3answers
172 views

$x^{y^z}$: is it $x^{(y^z)}$ or $(x^y)^z$?

Of the following, why is a usually considered true, and for what reason other than "tradition" and "more convenient"? a: ${x}^{y^z} = x^{(y^z)} \neq {(x^y)}^z$ b: ${x}^{y^z} = {(x^y)}^z \neq ...
3
votes
2answers
811 views

Obtaining the $\frac{1}{2\pi}$ factor in the Fourier transform

This MathWorld page gives this definition of a Fourier transform: $$F(k) = \int_{-\infty}^{\infty} f(x) e^{-2\pi i k x}dx.$$ But, I wish to speak in terms of linear frequency $\nu$ and time $t$ ...
3
votes
3answers
297 views

In what order are chained implications evaluated? (i.e. $a \implies b \implies c$)

Implication does not appear to be associative: a b c | (a -> b) -> c | a -> (b -> c) F T F | F | T F F F | F | T Is $a ...
3
votes
5answers
543 views

What type of input does trigonometric functions take in

I see in my Book that 45 deg is equivalent of π/4 . Ι also do the conversion if I simply convert degrees into radians like this 45* π/180 = π/4 radians and ...
3
votes
2answers
204 views

Can mathematical definitions of the form “P if Q” be interpreted as “P if and only if Q”? [duplicate]

Possible Duplicate: Alternative ways to say “if and only if”? So when I come across mathematical definitions like "A function is continuous if...."A space is compact ...
3
votes
1answer
158 views

Commas separating adjectives to describe mathematical objects.

Is there a convention which rules that in mathematics an object with some properties should be referred to as a "property 1 property 2 property 3 ...object" rather than a "property 1, property 2, ...
3
votes
2answers
48 views

Should “together with” be taken as slang for an n-tuple?

When an algebraic structure is defined, it is often defined as a set $S$ "along with"/"together with"/"having" operations $\circ_1, \circ_2, \ldots, \circ_n$, and "denoted" by $(S, \circ_1, \circ_2, ...
3
votes
1answer
263 views

Convention of digit grouping after decimal point

I read that different cultures have different ways of grouping digits before the decimal point for readability e.g. 1234567890 can be grouped as 1 234 567 890 (English), 12 3456 7890 (Chinese) or 1 23 ...
3
votes
2answers
127 views

Usage of $\cdot$ in calculus

I often find myself caught in the dilemma of whether or not to use the symbol $\cdot$ in calculus. Take for example, the chain rule: $$\frac{dy}{dx} = \frac{dy}{du}\cdot\frac{du}{dx}$$ Is the ...
3
votes
1answer
44 views

Commas between adjectives [duplicate]

Is it correct to use these expressions without commas (as written below)? A finite simple group. A finitely generated abelian group. A finitely generated torsion-free abelian group. A ...
3
votes
1answer
99 views

Category of topological pairs

Is there a standard abbreviation for the category of topological pairs? I have searched for it in vain.
2
votes
8answers
193 views

What is $2 - 1 + 1$? [duplicate]

$2-1+1$; a fairly straightforward question, but I (well, not me, but Henry Reich) found something strange. Most people would evaluate it as $2+(-1)+1 = 2$; however, this goes against the famed, and ...
2
votes
5answers
201 views

No difference between $0/0$ and $0^0$?

I have seen discussions about both $0/0$ and $0^0$ and they differ a bit in the way that most seem ok with calling $0/0$ "undefined", while the $0^0$ discussion still seems like a dispute. If this is ...
2
votes
1answer
55 views

Game Theory: players' gender convention?

What is the Game Theory convention of using gender terms (male/female) for the players? I found only one reference suggesting that odd-numbered players are male and even-numbered players are female. ...
2
votes
2answers
94 views

Is “iff” the same as equality if each member is a predicate?

"Iff" - if and only if ($\Leftrightarrow$ or $\leftrightarrow$, although the first usually carries a "meta" meaning, something that is not evaluated) - is used in $2x=3\Leftrightarrow x=\frac32$, ...
2
votes
1answer
221 views

How many solutions for $x^2 = 1$?

Let $F$ be an non-archimedean local field, let $o$ be its ring of integers, and let $p$ be the maximal ideal Is there a closed form for the cardinality $$ | \{ x \in o / p^N: x^2 = -1 \bmod p^N\} | ...
2
votes
1answer
80 views

Why is $m$ used as the variable for slope in slope-intercept form?

I was wondering if you could answer a question I have on slope intercept form of a linear equation. I know its $y=mx+b$, but why is it $mx+b$? Don't get me wrong. I know that $m$ is the slope and $b$ ...
2
votes
2answers
45 views

Why do we restrict the range of the inverse trig functions?

I understand why we restrict the domain, but why do we restrict the range? Why do we necessarily care so much for the inverse trig relations to be functions? Thanks!
2
votes
1answer
60 views

Define a domain filter of a function

Let $\mathbb{B}, \mathbb{V}$ two sets. I have defined a function $f: \mathbb{B} \rightarrow \mathbb{V}$. $\mathcal{P}(\mathbb{B})$ means the power set of $\mathbb{B}$, I am looking for a function ...
2
votes
2answers
62 views

Notation of an iterated function on 2 sets

Let $X$ and $C$ be two sets, I have defined an iterated function on them $f: X \times C \rightarrow X$. What interests me is the iterations of $f$ on an initial value $x \in X$, and a sequence ...
2
votes
1answer
166 views

Why is $0^0$ undefined? [duplicate]

Possible Duplicate: Zero to zero power I'm wondering why $0^0$ is considered undefined. Why isn't 1 considered a valid solution? Considering $0^0 = 1$ seems reasonable to me for two ...
2
votes
2answers
390 views

Why is the range of arctan $[ -\frac{\pi}{2} , \frac{\pi}{2}]$?

I've been taught in school and it says on Wikipedia that the range of arctan is $[ -\frac{\pi}{2} , \frac{\pi}{2} ]$. Why isn't it $[0,\pi]$ ?
2
votes
1answer
35 views

Semidirect product notation convention

I was taught that if $G \simeq H \ltimes K$, then (by convention) $H$ is the subgroup that is normal, but I see on Wikipedia and elsewhere that other people use the convention that the above indicates ...
2
votes
2answers
90 views

why function argument is on right side $f(x)$ rather than on left side as $xf$

Is there an advantage for writing function arguments on the right side as $f(x)$ rather than on the left side as $xf$? The latter looks more natural if we think about it in diagram as $domain ...
2
votes
2answers
70 views

What is a vector with a single non-zero component called?

What do you call a vector with a single component, e.g. $[a, 0, ..., 0]$ or $[0, b, 0, ..., 0]$, where $a, b$ are any non-zero number? I'd like the language to differentiate from a vector with ...
2
votes
1answer
43 views

What's the name of each pseudo-rectangle in a spherical surface?

Consider the common surface of a spherical segment crossed with a spherical wedge. This produces a pseudo-rectangle in the sphere surface, and a perfect rectangle in a mercator projection. What's the ...
2
votes
2answers
48 views

Name of region defined by two points in higher dimensions

Perhaps this isn't the best place to ask, but I'm trying to figure out the proper name for the region that exists in-between two points in higher dimensions. Under certain conditions, n-cube or ...
2
votes
2answers
6k views

Square root of a number squared is equal to the absolute value of that number [duplicate]

Possible Duplicate: Significance of $\displaystyle\sqrt[n]{a^n} $? The square root of a number squared is equal to the absolute value of that number. Why is $\sqrt{x^2} = |x|$? Why not just ...
2
votes
3answers
440 views

Why are there two possible triangles when given SAS?

I gave my trigonometry students the following example: Solve $\triangle ABC\ $ , where AC=0.923, AB=.387, and $\measuredangle A\ = 43.33^\circ\ $. First I found BC using the law of cosines, then I ...
1
vote
2answers
43 views

Notation of functions which get a element of a pair

Given a pair $(a, b) \in A \times B$, I would like to know how to write the functions which get the first element and the second element of the pair... In a programming language, one can write ...