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I know personally made notations are generally a bad thing, but I've not seen any reason to stop using the notation I've made, and it feels more natural to use. Now, this my seem like a biased question to begin with, but my hoping was that there would be a natural reasoning why we don't already use notation like this to begin with.

Now let me get to the point: angled brackets for parameter: $\langle*\rangle$. I can't seem to understand the point of using $(*)$ to denote a parameter, when this can too easily be misunderstood with multiplication. To show an example of how this notation makes things clearer, let me demonstrate with the classical notation of $\ln(x+y)^2$. With classical notation, it's hard to distinguish this from either $\ln((x+y)^2)$ and $\ln(x+y)\times\ln(x+y)$. Using $\langle*\rangle$ for the parameter, would make this simple to distinguish, as the latter would be written $\ln\langle x+y\rangle^2$. The first would be written as $\ln\langle(x+y)^2\rangle$, or simply $\ln(x+y)^2$, as the angled brackets would be redundant. Some other examples:

$$ \begin{align} \text{Normal notation:} \\ \sin(x)^2 &\stackrel{?}{=} \sin x^2 \textbf{ or } \sin x\times\sin x = \sin^2x\\[1em] \text{My notation:} \\ \sin(x)^2 &= \sin x^2 \text{, while }\\ \sin\langle x\rangle^2 &= \sin x \times\sin x = \sin^2 x \\[2em] \text{Normal notation:} \\ f(a+b) &\stackrel{?}{=} f\times a+f\times b\textbf{ or } \text{a function of }a+b\\[1em] \text{My notation:} \\ f(a+b) &= f\times a+f\times b \text{, while }\\ f\langle a+b\rangle &= \text{a function of }a+b \end{align} $$

This notation also allows for one to postfix the function, as $\langle *\rangle$ would always be the parameter of some function $f$.

$$ \begin{align} f\big\langle g \langle x \rangle \big\rangle = \big\langle\langle x \rangle g\big\rangle f \end{align} $$

This allows $n!$ to be written as $!\langle n\rangle$, or $|x|$ as $||\langle x\rangle$, so functions and parameters can easily be distinguished. Now I figure that if this notation was useful, we'd use it already. Yet I haven't found any drawbacks by using this notation, as the angled bracked $\langle \rangle$ are somewhat uncommon in mathematics, yet are really easy to draw and distinguish from normal brackets $( )$.

So the question really is; why do we use the same brackets for multiplication as we do for parameters, and would it be beneficial to change this notation?

Frank Vel
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    If for no other reason: tradition. Would it be beneficial to change? No, it would cause too much upheaval. The current notation just isn't bad enough to justify the disruption a change like that would cause, even if it would otherwise have some minor benefits. You need a huge benefit to justify that sort of change. – Daniel Fischer Oct 10 '14 at 12:29
  • Another used placeholder is a dot, like in $|.|$ or $<.,.>$. – mvw Oct 10 '14 at 12:32
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    This new notation reminds me of some mathematics-oriented programming languages, namely Wolfram language (the one used in Wolfram Mathematica). There f(x) is equivalent to f*x while f[x] is equivalent to f@x, i.e. f is a function applied to x. – Ruslan Oct 10 '14 at 12:34
  • To add another reason: the speed of typing the symbols () is much greater than \langle \rangle. – TZakrevskiy Oct 10 '14 at 12:35
  • And another one - what do you propose to do with the cases where we use $\langle \rangle$ in a meaningful way? Like, for example, distributions? – TZakrevskiy Oct 10 '14 at 12:36
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    Or inner products. Notation is overloaded all the time. – Alan Oct 10 '14 at 12:39
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    Actually one should not write $\ln(x+y)^2$ to mean $\ln(x+y)\cdot\ln(x+y)$. One should write $\ln^2(x+y)$ or $(\ln(x+y))^2$ and all the problems are solved. – marco trevi Oct 10 '14 at 12:42
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    Notation is a language, and making up your own notation is making up your own language - it makes it hard for you to communicate with people. Since the purpose of language is communication, making up your own language is effectively pointless. – Thomas Andrews Oct 10 '14 at 12:42
  • @DanielFischer Fischer: To me it doesn't require a lot of thought to change, just adding an angle to standard brackets doesn't make it unreadable and is simple to understand. – Frank Vel Oct 10 '14 at 12:43
  • @ThomasAndrews I think that making up notation is useful when understanding things, i.e. when communicating to yourself :) – marco trevi Oct 10 '14 at 12:44
  • @TZakrevskiy: That is one of my legitimate concerns, but I've never seen the angled brackets being used too much. In that case $\langle \rangle$ could be considered the function? – Frank Vel Oct 10 '14 at 12:46
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    @marcotrevi: $\ln^2(x)$ would mean $\ln(\ln(x))$ afaik? – Frank Vel Oct 10 '14 at 12:47
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    No, $\ln^2(x)=\ln(x)\cdot\ln(x)$, while $\ln(\ln(x))$ means $\ln(\ln(x))$ – marco trevi Oct 10 '14 at 12:48
  • @fvel, for distributions $\langle u,\phi\rangle$ is the action of a distribution $u$ on a test function $\phi$. This is to distinguish from inner product, which often used at the same time. – TZakrevskiy Oct 10 '14 at 12:49
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    @marcotrevi can I have some source for that? I've never seen $f^2(x)$ mean anything other than $f(f(x))$ for other than $\sin^2(x) = \sin(x)\times \sin(x)$ – Frank Vel Oct 10 '14 at 12:51
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    so, $(x+1)^2$ should mean $x+2$ in your opinion? – marco trevi Oct 10 '14 at 12:54
  • @TZakrevskiy: I'm not familiar with this use, and I can see why it would be a problem. Would it be possible to write $\langle\rangle\langle u, \phi\rangle$ ? Or just let $\langle u, \phi \rangle$ without a function be defined to be the distibution of $u$ in that case, as $u$ and $\phi$ already are parameters? – Frank Vel Oct 10 '14 at 12:57
  • @marcotrevi as stated I wish for $()$ to be considered multiplication-brackets. $(x+1)^2$ would mean $(x+1)(x+1)$, whilst $\langle x+1 \rangle^2$ would mean that $x+1$ are the parameters for some function $^2$ – Frank Vel Oct 10 '14 at 13:01
  • Cannot edit last comment: $(x+1)^2 = (x+1)\times(x+1) = x^2+2x+1$ while $\langle x+1\rangle^2 = \langle x+1\rangle\times\langle x+1\rangle$. – Frank Vel Oct 10 '14 at 17:11

2 Answers2

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I never ever write $\sin x^2$ when I want to write $\sin x \cdot \sin x$ - absolutely don't do this - as it will definitely cause massive confusion. Normally angular brackets are used to define the component form of a vector, for example $$\vec{F}(x,y,z)=\langle P,Q,R \rangle \equiv P\vec{i}+Q\vec{j}+R\vec{k}$$ But as long as you state what you mean in the text, it should be ok.

S. Feinberg
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Okay I think I've got enough responses to answer this question myself. I might add more to this list if more things come up, or modify it if what is written is incorrect. I've tried not to be too biased in this list, although I must admit that I'll start using this notation.

Pros

  • Already in use

    The Wolfram language already uses notation like this to distinguish multiplication $f(x)$ from applying a function $f[x]$, although this uses square brackets $[]$ instead of angled brackets $\langle\rangle$. If programming languages finds it sensible to use similar notation, it could work in mathematics too.

  • Information is added, not removed

    If you've already written a complicated expression like $\exp(h(f(a+b)^2-f(a-c)))$ where $h$ is a constant and $f$ is a function and $a,b,c$ are parameters, this can be difficult to read. Writing $\exp\langle h(f\langle a+b\rangle^2-f\langle a-c\rangle)\rangle$ might make it clear what's parameters, and what's to be multiplied.

  • Easy to learn

    This would be a lot easier to learn than notations like $\binom{n}{k}$ or $n!$, because the notation is already similar to the one in use.

Cons

  • Existing functions:

    There exist functions using $\langle \rangle$ notation, such as $\langle u, \phi\rangle$, and these could be difficult to incorporate in this new system, because there would be no symbol for the function. Although the lack of a symbol could be seen as a symbol.

  • Speed of typing

    It would be slightly more inconvenient to use $\langle \rangle$ instead of $()$ as it would require one to write more symbols, since "\langle" is longer than "(", although this problem would only occur when writing $\LaTeX$, not on paper, and could be incorporated as a keyboard shortcut.

  • Traditions

    It would require some learning to be accustomed to this new notation, and would not be worth the hassle to teach people. Its benefits wouldn't outweigh its disadvantages.

Frank Vel
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