Why start at x, y, z - User contributions [en-gb]

Juxtaposition means combine in the obvious way

2026-05-22T13:05:18Z

Christian Lawson-Perfect:

[[Category:Unspoken conventions]]
[[Category:Ambiguities]]

In an expression, putting two things immediately next to each other usually means that they should be combined in some way. It's usually implicit that the combination operation should be clear from the context.

==Examples==

* [[Missing multiplication symbol|Multiplication]]: $ab = a \times b$.
* Function composition: $fg(x) = f \circ g(x) = f(g(x))$.
* [[Function application without parentheses|Function application]]: $\sin x$.
* Group operation: when $x,y \in G = (X,\star)$, $xy = x \star y$.
* A linear transformation: $\mathrm{A}\mathbf{v}$. (I've never seen $\mathrm{A} \times \mathbf{v}$ or $\mathrm{A} \cdot \mathbf{v}$ for matrix-vector product)

==Exceptions==

* [[Matrix indices|Matrix indices written without a comma]].

==Unicode==

There are Unicode symbols U+2062 INVISIBLE TIMES and U+2061 FUNCTION APPLICATION, both invisible. Christian Lawson-Perfect claims he can tell when you don't use them<ref>[https://somethingorotherwhatever.com/twitter-archive/status/1321409295473352705/ Tweet by Christian Lawson-Perfect]</ref>.

==References==
<references/>

Juxtaposition means combine in the obvious way

2026-05-22T13:04:53Z

Christian Lawson-Perfect:

[[Category:Unspoken conventions]]
[[Category:Ambiguities]]

In an expression, putting two things immediately next to each other usually means that they should be combined in some way. It's usually implicit that the combination operation should be clear from the context.

==Examples==

* [[Missing multiplication symbol|Multiplication]]: $ab = a \times b$.
* Function composition: $fg(x) = f \circ g(x) = f(g(x))$.
* [[Function application without parentheses|Function application]]: $\sin x$.
* Group operation: when $x,y \in G = (X,\star)$, $xy = x \star y$.
* A linear transformation: $\mathrm{A}\mathbf{v}$. (I've never seen $\mathrm{A} \times \mathbf{v}$ or $\mathrm{A} \cdot \mathbf{v}$ for matrix-vector product)

==Exceptions==

* [[Matrix indices|Matrix indices written without a comma]].

==Unicode==

There are Unicode symbols U+2062 INVISIBLE TIMES and U+2061 FUNCTION APPLICATION, both invisible. Christian Lawson-Perfect claims he can tell when you don't use them<ref>https://somethingorotherwhatever.com/twitter-archive/status/1321409295473352705/ Tweet by Christian Lawson-Perfect</ref>.

==References==
<references/>

Juxtaposition means combine in the obvious way

2026-05-22T13:04:13Z

Christian Lawson-Perfect:

[[Category:Unspoken conventions]]
[[Category:Ambiguities]]

In an expression, putting two things immediately next to each other usually means that they should be combined in some way. It's usually implicit that the combination operation should be clear from the context.

==Examples==

* [[Missing multiplication symbol|Multiplication]]: $ab = a \times b$.
* Function composition: $fg(x) = f \circ g(x) = f(g(x))$.
* [[Function application without parentheses|Function application]]: $\sin x$.
* Group operation: when $x,y \in G = (X,\star)$, $xy = x \star y$.
* A linear transformation: $\mathrm{A}\mathbf{v}$. (I've never seen $\mathrm{A} \times \mathbf{v}$ or $\mathrm{A} \cdot \mathbf{v}$ for matrix-vector product)

==Exceptions==

* [[Matrix indices|Matrix indices written without a comma]].

==Unicode==

There are Unicode symbols U+2062 INVISIBLE TIMES and U+2061 FUNCTION APPLICATION, both invisible. <ref>https://somethingorotherwhatever.com/twitter-archive/status/1321409295473352705/ Christian Lawson-Perfect claims he can tell when you don't use them</ref>.

==References==
<references/>

Juxtaposition means combine in the obvious way

2026-05-22T13:03:57Z

Christian Lawson-Perfect:

[[Category:Unspoken conventions]]
[[Category:Ambiguities]]

In an expression, putting two things immediately next to each other usually means that they should be combined in some way. It's usually implicit that the combination operation should be clear from the context.

==Examples==

* [[Missing multiplication symbol|Multiplication]]: $ab = a \times b$.
* Function composition: $fg(x) = f \circ g(x) = f(g(x))$.
* [[Function application without parentheses|Function application]]: $\sin x$.
* Group operation: when $x,y \in G = (X,\star)$, $xy = x \star y$.
* A linear transformation: $\mathrm{A}\mathbf{v}$. (I've never seen $\mathrm{A} \times \mathbf{v}$ or $\mathrm{A} \cdot \mathbf{v}$ for matrix-vector product)

==Exceptions==

* [[Matrix indices|Matrix indices written without a comma]].

==Unicode==

There are Unicode symbols U+2062 INVISIBLE TIMES and U+2061 FUNCTION APPLICATION, both invisible. <ref>https://somethingorotherwhatever.com/twitter-archive/status/1321409295473352705/ Christian Lawson-Perfect claims he can tell when you don't use them</ref>.

Missing multiplication symbol

2026-05-22T13:01:46Z

Christian Lawson-Perfect:

[[Category:Ambiguities]]

It's common to omit a multiplication symbol:

$ab = a \times b$

But sometimes it's not as clear:

Does $ a(b+1) = a \times (b+1)$, or is $a$ a function?<ref>[https://twitter.com/christianp/status/798843905231888385 Tweet by Christian Lawson-Perfect]</ref>

When writing a division on one line, does an implied multiplication bind more tightly than an explicit one?<ref>[https://twitter.com/christianp/status/1320650593241866241 Twitter thread by Christian Lawson-Perfect]</ref>

Is $a/bc$ equivalent to $\frac{a}{bc}$ or $\frac{a}{b}c$?

There seems to be an unwritten rule "juxtaposition is stickier". (See [[Juxtaposition means combine in the obvious way]])

In linear notation, you get the same problem: does <code>e^xy^2</code> mean $e^{x y^2}$ or $e^x y^2$? Here, the "juxtaposition is stickier" rule doesn't feel right.

But that might not apply when there are numbers involved: almost everyone would interpret $2/3x$ as $\frac{2}{3}x$ instead of $\frac{2}{3x}$

The "juxtaposition is stickier" rule only seems to break ties, not override the normal [[order of operations]]:

\[ ab^2 = a \times (b^2) \]

Sometimes the ambiguity comes from mistaking a function for an operation:

\[ (a+b) \Phi (a+b)\]

which can be viewed as either $ (a+b)\cdot \Phi(a+b)$, or $\Phi$ as binary addition-like operation, similar to $ (a+b)\oplus (a+b)$.<ref>Igor Pak, [https://scholarship.claremont.edu/jhm/vol8/iss1/14/ How to Write a Clear Math Paper: Some 21st Century Tips]</ref>

Jim Simons reckons we should give up on implicit multiplication altogether.<ref>The Times, They Are A-Changin', Jim Simons, Mathematics in School, November 2020.</ref>

There is a unicode symbol U+2062 INVISIBLE TIMES.

==References==
<references/>

Missing multiplication symbol

2026-01-06T14:31:07Z

Christian Lawson-Perfect:

Ways of writing numbers

2024-08-23T14:16:51Z

Christian Lawson-Perfect:

[[Category:References]]
[[Category:Needs filling in]]

<ref>[https://twitter.com/christianp/status/1240294851641516036 Thread started by Christian Lawson-Perfect: " want as many different conventions for writing numbers as you can give me."]</ref>

<ref>[https://mathstodon.xyz/@christianp/113011654970305019 Repost of the Twitter thread on mathstodon.xyz"]</ref>

Category:Local variations

2022-10-20T04:55:54Z

Christian Lawson-Perfect:

This category collects conventions in notation that are particular to a place or group of people.

https://twitter.com/xkcd/status/1582839215632527363

When the left and right delimiter are the same symbol

2022-10-18T12:08:21Z

Christian Lawson-Perfect:

[[Category:Ambiguities]]

When the left and right delimiter are the same symbol, for example a vertical line, there can be more than one way of interpreting an expression with several sets of that delimiter.

For example, when a vertical line is used to represent the absolute value (or [[Modulus|modulus]]) of a number, the following expression has at least two plausible interpretations, because [[Juxtaposition means combine in the obvious way]]

\[ |x|y|z| \]

If we write "the absolute value of $x$" as "$\operatorname{abs}(x)$", then the expression above could be interpreted as:

* $\operatorname{abs}(x) \cdot y \cdot \operatorname{abs}(z)$
* $\operatorname{abs}(x \cdot \operatorname{abs}(y) \cdot z)$

When the left and right delimiter are the same symbol

2022-10-18T12:08:03Z

Christian Lawson-Perfect:

[[Category:Ambiguities]]

When the left and right delimiter are the same symbol, for example a vertical line, there can be more than one way of interpreting an expression with several sets of that delimiter.

For example, when a vertical line is used to represent the absolute value (or [[Modulus|modulus]]) of a number, the following expression has at least two plausible interpretations, because [[Juxtaposition means combine in the obvious way]]

\[ |x|y|z| \]

If we write "the absolute value of $x$" as "$\operatorname{abs}(x)$", then the expression above could be interpreted as:

* $\operatorname{abs}(x) \cdot y \cdot \operatorname{abs}(z)$
* $\operatorname{abs}(x \cdot \operatorname{abs}(y) \cdot z)$

When the left and right delimiter are the same symbol

2022-10-18T12:07:50Z

Christian Lawson-Perfect: Created page with "Category:Ambiguities When the left and right delimiter are the same symbol, for example a vertical line, there can be more than one way of interpreting an expression with..."

[[Category:Ambiguities]]

When the left and right delimiter are the same symbol, for example a vertical line, there can be more than one way of interpreting an expression with several sets of that delimiter.

For example, when a vertical line is used to represent the absolute value (or [[Modulus|modulus]]) of a number, the following expression has at least two plausible interpretations, because [[Juxtaposition means combine in the obvious way]]

\[ |x|y|z| \]

If we write "the absolute value of $x$" as "$\operatorname{abs}(x)$", then the expression above could be interpreted as:

* $\operatorname{abs}(x) \cdot y \cdot \operatorname{abs}(z)$
* $\operatorname{abs}(x \cdot \operatorname{abs}(y) \cdot z)$

Arabic percent sign

2022-10-12T05:01:06Z

Christian Lawson-Perfect: Created page with "Unicode has a specifically Arabic percent sign symbol: ٪ Category:Local variations"

Unicode has a specifically Arabic percent sign symbol: ٪

[[Category:Local variations]]

Quotes

2022-04-18T16:31:39Z

Christian Lawson-Perfect:

[[Category:References]]

<blockquote>Symboles are poor unhandsome (though necessary) scaffolds of demonstration; and ought no more to appear in publique, then the most deformed necessary business which you do in your chambers. (Hobbes 1656)</blockquote>

<hr>

https://twitter.com/DavidKButlerUoA/status/1512935483583590400

<hr>

https://twitter.com/AndresECaicedo1/status/1503074377910345728

<hr>

https://twitter.com/AndresECaicedo1/status/1500650582905954308

<hr>

https://twitter.com/matthematician/status/1516047877356802052

Quotes

2022-04-10T05:28:47Z

Christian Lawson-Perfect:

Quotes

2022-04-10T05:21:57Z

Christian Lawson-Perfect:

Quotes

2022-04-10T05:17:58Z

Christian Lawson-Perfect:

Quotes

2022-04-10T05:17:42Z

Christian Lawson-Perfect:

Elliptic integrals

2021-11-20T13:02:26Z

Christian Lawson-Perfect: Created page with "<ref>[https://cybre.space/@apocheir/107237017773898587 Toot by @apocheir@cybre.space]</ref> Category:Needs filling in"

<ref>[https://cybre.space/@apocheir/107237017773898587 Toot by @apocheir@cybre.space]</ref>

[[Category:Needs filling in]]

Separating arguments of a function

2021-11-20T13:00:10Z

Christian Lawson-Perfect:

In English, a comma normally separates components of a bracketed list, such as the arguments of a function.

For example, $f(x,y)$ is the application of a function $f$ to two arguments, $x$ and $y$.

When the convention for number notation is to use a comma as the decimal separator, this can lead to an ambiguity when the arguments are numbers.<ref>[https://mathstodon.xyz/@christianp/107282002688892264 Toot by Christian Lawson-Perfect], [https://twitter.com/christianp/status/1460282403495284740 Tweet by Christian Lawson-Perfect]</ref>

For example, is $f(1,2)$ the application of a function of one argument, or two?

Instead, a common convention is to use a semicolon as the item separator.

For example, $f(1;2)$ is unambiguously a function of two arguments.

Others use spacing to separate items, in addition to a comma, such as $f(1,\, 2)$. (but [[Space is significant]]!)

[[Category:Ambiguities]]
[[Category:Local variations]]

Separating arguments of a function

2021-11-20T12:54:38Z

Christian Lawson-Perfect:

In English, a comma normally separates components of a bracketed list, such as the arguments of a function.

For example, $f(x,y)$ is the application of a function $f$ to two arguments, $x$ and $y$.

When the convention for number notation is to use a comma as the decimal separator, this can lead to an ambiguity when the arguments are numbers.

For example, is $f(1,2)$ the application of a function of one argument, or two?

Instead, a common convention is to use a semicolon as the item separator.

For example, $f(1;2)$ is unambiguously a function of two arguments.

Others use spacing to separate items, in addition to a comma, such as $f(1,\, 2)$. (but [[Space is significant]]!)

[[Category:Ambiguities]]
[[Category:Local variations]]

Separating arguments of a function

2021-11-20T12:54:08Z

Christian Lawson-Perfect: Created page with "In English, a comma normally separates components of a bracketed list, such as the arguments of a function. For example, $f(x,y)$ is the application of a function $f$ to..."

In English, a comma normally separates components of a bracketed list, such as the arguments of a function.

For example, $f(x,y)$ is the application of a function $f$ to two arguments, $x$ and $y$.

When the convention for number notation is to use a comma as the decimal separator, this can lead to an ambiguity when the arguments are numbers.

For example, is $f(1,2)$ the application of a function of one argument, or two?

Instead, a common convention is to use a semicolon as the item separator.

For example, $f(1;2)$ is unambiguously a function of two arguments.

Others use spacing to separate items, in addition to a comma, such as $f(1,\, 2)$. (but [[Space is significant]]!)

Series that are neither convergent nor divergent

2021-11-20T12:46:57Z

Christian Lawson-Perfect:

According to Adam Atkinson, in Italy a series that isn't convergent and doesn't tend to $\pm \infty$ is "indeterminate", not "divergent".

[[Category:Local variations]]
[[Category:Conflicting definitions]]

Series that are neither convergent nor divergent

2021-11-20T11:46:47Z

Christian Lawson-Perfect: Created page with "According to Adam Atkinson, in Italy a series that isn't convergent and doesn't tend to $\pm \infty$ is "indeterminate", not "divergent"."

According to Adam Atkinson, in Italy a series that isn't convergent and doesn't tend to $\pm \infty$ is "indeterminate", not "divergent".

Symbols with no standard pronunciation

2021-11-07T16:10:24Z

Christian Lawson-Perfect: Created page with "== Greek letters == In English, there are multiple widely-accepted ways of pronouncing the names of Greek letters.<ref>[https://english.stackexchange.com/questions/11363/why-..."

== Greek letters ==

In English, there are multiple widely-accepted ways of pronouncing the names of Greek letters.<ref>[https://english.stackexchange.com/questions/11363/why-are-greek-letters-pronounced-incorrectly-in-scientific-english English language & usage StackExchange question - "Why are Greek letters pronounced incorrectly in scientific English?"]</ref>

For $\beta$, in Britain "bee-ta" is most common, while in the USA "bay-ta" is more common. Similar for $\zeta$.

== 0 ==

The symbol 0 is spoken in many different ways in English:

* zero
* nought
* naught/nowt
* null
* nil
* oh
* cipher

[[Category:Spoken language]]
[[Category:Local variations]]

Functions with no standard pronunciation

2021-11-07T16:02:48Z

Christian Lawson-Perfect:

There are many mathematical functions with a single widely-agreed written form, but with multiple widely-used spoken forms.

== Hyperbolic functions ==

Common ways of saying $\sinh$ include "shine", "sine-ch", "sine-aitch", and "hyperbolic sine"<ref>[https://twitter.com/mrsouthernmaths/status/1457289072922148864 Twitter poll by Rob Southern - "how do you pronounce sinh?"]</ref>.

This applies similarly to $\tanh$, but $\cosh$ can be read phonetically in English.

== Other examples ==

* $\ln$ - "lunn", "ell enn", "natural log(arithm)".
* $\operatorname{ercf}$

[[Category:Local variations]]
[[Category:Spoken language]]

Category:Spoken language

2021-11-07T16:02:23Z

Christian Lawson-Perfect:

This category collects things to do with the way mathematics is spoken out loud.

[[Category:Language]]

Functions with no standard pronunciation

2021-11-07T16:02:09Z

Christian Lawson-Perfect: Created page with "There are many mathematical functions with a single widely-agreed written form, but with multiple widely-used spoken forms. == Hyperbolic functions == Common ways of saying..."

There are many mathematical functions with a single widely-agreed written form, but with multiple widely-used spoken forms.

== Hyperbolic functions ==

Common ways of saying $\sinh$ include "shine", "sine-ch", "sine-H", and "hyperbolic sine"<ref>[https://twitter.com/mrsouthernmaths/status/1457289072922148864 Twitter poll by Rob Southern - "how do you pronounce sinh?"]</ref>.

This applies similarly to $\tanh$, but $\cosh$ can be read phonetically in English.

== Other examples ==

* $\ln$ - "lunn", "ell enn", "natural log(arithm)".
* $\operatorname{ercf}$

[[Category:Local variations]]
[[Category:Spoken language]]

Category:Spoken language

2021-11-07T15:56:16Z

Christian Lawson-Perfect: Created page with "This category collects things to do with the way mathematics is spoken out loud."

This category collects things to do with the way mathematics is spoken out loud.

Lack of brackets in spoken language

2021-11-07T15:55:46Z

Christian Lawson-Perfect:

There are problems similar to those related to [[order of operations]] misunderstandings causes by the lack of brackets in language.

For example, if trying to describe $3^{2x}$, you might say "three to the power of two times x". This could, however, also be interpreted as $3^{2}x$. One common way to reduce the ambiguity is to pause and speed up, ie say "three to the power of [''pause''] two-times-x".

Another example are sentences "17 is a factor of 6 more than 15" and "7 is a factor of 6 more than 15". Both can be interpreted as correct under different readings: 17 is (a factor of 6) more than 15, 7 is a factor of (6 more than 15). The first of these could be disambiguated by saying "17 is 15 plus a factor of 6", but there is no obvious unambiguous candidate for the second.

Matthew Scroggs finds this a particular challenge when writing clues for the [https://chalkdustmagazine.com/regulars/crossnumber Chalkdust crossnumber].

[https://aperiodical.com/2013/02/all-squared-number-1-maths-out-loud/ The first episode of the All Squared podcast] is about spoken mathematics.

[[Category:Ambiguities]]
[[Category:Spoken language]]

References

2021-10-16T14:42:53Z

Christian Lawson-Perfect:

[[Category:References]]

[http://www.maths.ed.ac.uk/~aar/papers/cajorinot.pdf A History of Mathematical Notations] by Florian Cajori.

Jeff Miller maintains some pages on the earliest known uses of [https://jeff560.tripod.com/mathword.html mathematical words] and [https://jeff560.tripod.com/mathsym.html mathematical symbols], as well as [https://jeff560.tripod.com/ambiguities.html a list of ambiguities encountered at high school].

[[User:Christian Lawson-Perfect | Christian Lawson-Perfect]] collects [https://read.somethingorotherwhatever.com/collection/notation-and-conventions links to papers, books and other stuff about notation and conventions].

[https://math.vanderbilt.edu/schectex/commerrs/ Common Errors in College Mathematics]

[https://dlmf.nist.gov/front/introduction#notations The Digital Library of Mathematical Functions' notation section]

[https://math.stackexchange.com/questions/1024280/most-ambiguous-and-inconsistent-phrases-and-notations-in-maths Most ambiguous and inconsistent phrases and notations in maths] on math.stackexchange

[https://www.theallusionist.org/numbers Allusionist podcast 140: "Numbers"]

Space is significant

2021-10-04T10:15:39Z

Christian Lawson-Perfect:

[[Category:Ambiguities]]
[[Category:Handwriting]]

Empty space is often significant in mathematical notation, but it's easy to misinterpret, and hard to produce accurately.

TeX applies different amounts of spacing around symbols based on context. The ubiquity of TeX means that typeset expressions without this spacing look odd.

There's some evidence that people learning algebra use the spacing of symbols to deduce rules.<ref>[https://www.jstor.org/stable/30034809 Visual salience of Algebraic Transformations], David Kirshner and Thomas Awtry, Journal for Research in Mathematics Education.</ref>

Producing reliable spacing in handwriting is hard.

==Examples==

Entries in a matrix are separated by empty space<ref>[https://twitter.com/christianp/status/1444937081915068416 Tweet by Christian Lawson-Perfect]</ref>:

\[ \begin{pmatrix} 1 & - 2 \\ 3 & - 4 \end{pmatrix} \]

\[ \begin{pmatrix} 1-2 \\ 3 - 4 \end{pmatrix} \]

<hr/>

Statements about a line are usually separated from the main expression by a large space.

\[ 4 \equiv 1 \mod 3 \]

versus

\[ 4 \equiv 1 \bmod 3\]

<hr/>

Juxtaposing two fractions:

\[ \frac{1}{2} \frac{-3}{4} \]

\[ \frac{1-3}{2\, 4} \]

==References==

<references/>

Space is significant

2021-10-04T08:51:18Z

Christian Lawson-Perfect: /* Examples */

[[Category:Ambiguities]]

Empty space is often significant in mathematical notation, but it's easy to misinterpret, and hard to produce accurately.

TeX applies different amounts of spacing around symbols based on context. The ubiquity of TeX means that typeset expressions without this spacing look odd.

There's some evidence that people learning algebra use the spacing of symbols to deduce rules.<ref>[https://www.jstor.org/stable/30034809 Visual salience of Algebraic Transformations], David Kirshner and Thomas Awtry, Journal for Research in Mathematics Education.</ref>

==Examples==

Entries in a matrix are separated by empty space<ref>[https://twitter.com/christianp/status/1444937081915068416 Tweet by Christian Lawson-Perfect]</ref>:

\[ \begin{pmatrix} 1 & - 2 \\ 3 & - 4 \end{pmatrix} \]

\[ \begin{pmatrix} 1-2 \\ 3 - 4 \end{pmatrix} \]

<hr/>

Statements about a line are usually separated from the main expression by a large space.

\[ 4 \equiv 1 \mod 3 \]

versus

\[ 4 \equiv 1 \bmod 3\]

<hr/>

Juxtaposing two fractions:

\[ \frac{1}{2} \frac{-3}{4} \]

\[ \frac{1-3}{2\, 4} \]

==References==

<references/>

Space is significant

2021-10-04T08:41:00Z

Christian Lawson-Perfect: /* Examples */

[[Category:Ambiguities]]

Empty space is often significant in mathematical notation, but it's easy to misinterpret, and hard to produce accurately.

TeX applies different amounts of spacing around symbols based on context. The ubiquity of TeX means that typeset expressions without this spacing look odd.

There's some evidence that people learning algebra use the spacing of symbols to deduce rules.<ref>[https://www.jstor.org/stable/30034809 Visual salience of Algebraic Transformations], David Kirshner and Thomas Awtry, Journal for Research in Mathematics Education.</ref>

==Examples==

Entries in a matrix are separated by empty space<ref>[https://twitter.com/christianp/status/1444937081915068416 Tweet by Christian Lawson-Perfect]</ref>:

\[ \begin{pmatrix} 1 & - 2 \\ 3 & - 4 \end{pmatrix} \]

\[ \begin{pmatrix} 1-2 \\ 3 - 4 \end{pmatrix} \]

<hr/>

Statements about a line are usually separated from the main expression by a large space.

\[ 4=1 \mod 3 \]

versus

\[ 4=1 \bmod 3\]

<hr/>

Juxtaposing two fractions:

\[ \frac{1}{2} \frac{-3}{4} \]

\[ \frac{1-3}{2\, 4} \]

==References==

<references/>

Order of operations

2021-09-16T11:49:56Z

Christian Lawson-Perfect:

There are quite a few mnemonics for the order of operations. A common one in the UK is BODMAS:

* Brackets
* Orders
* Division
* Multiplication
* Addition
* Subtraction

Elsewhere, PEMDAS is popular.

But division and multiplication have equal precedence, and so do addition and subtraction. A common convention is that operations with equal precedence are evaluated from left to right.

This leads to all sorts of misunderstandings.

$ 8 \div 2(1+3) = 16 $ or $ 1 $

Some people think that the presence or omission of a multiplication symbol in the above expression is important: [[Missing multiplication symbol|implicit multiplication]] might bind more tightly than the division symbol.

Several ways of resolving the ambiguity have been suggested, but all the ones [[User:Christian Lawson-Perfect | I've]] seen introduce other problems.

==Suggested resolutions==

'''Reverse Polish Notation''': The expression $(x-2)(x-1)$ would be written instead $\times \, - \, x \, 2 \, - \, x \, 1$, or something like that. There's no need for brackets or operator precedence, but it is hard to see at a glance what each operator applies to.

'''Add brackets''': around everything??

'''Make up a new rule''': At MathsJam Gathering 2020, Christian Lawson-Perfect suggested adding a rule "M before D except after 3". So $6 \div 2 \times 3 = 1$, but $3 \div 2 \times 4 = 6$.

==References==

Christian Lawson-Perfect has made a tool called [https://www.checkmyworking.com/misc/samdob/ SAMDOB] which lets you make up your own mnemonic and see how an expression is evaluated.

Colin Beveridge dubbed viral "puzzles" related to misunderstanding the order of operations [https://www.flyingcoloursmaths.co.uk/new-years-resolution-genius-sic/ fake maths].

Adam Townsend and Matthew Scroggs proposed using [https://chalkdustmagazine.com/blog/medusa-new-bodmas/ MEDUSA] instead of BODMAS to reduce ambiguities caused by division being done before multiplication or subtraction being doing before addition.

Kyle D. Evans' book, [https://atlantic-books.co.uk/book/maths-tricks-to-blow-your-mind/ Maths Tricks to Blow Your Mind], collects many examples of viral order of operations problems in Chapter 4.

[[Category:Ambiguities]]

Order of operations

2021-09-16T11:47:47Z

Christian Lawson-Perfect:

There are quite a few mnemonics for the order of operations. A common one in the UK is BODMAS:

* Brackets
* Orders
* Division
* Multiplication
* Addition
* Subtraction

Elsewhere, PEMDAS is popular.

But division and multiplication have equal precedence, and so do addition and subtraction. A common convention is that operations with equal precedence are evaluated from left to right.

This leads to all sorts of misunderstandings.

$ 8 \div 2(1+3) = 16 $ or $ 1 $

Some people think that the presence or omission of a multiplication symbol in the above expression is important: [[Missing multiplication symbol|implicit multiplication]] might bind more tightly than the division symbol.

Several ways of resolving the ambiguity have been suggested, but all the ones [[User:Christian Lawson-Perfect | I've]] seen introduce other problems.

==Suggested resolutions==

'''Reverse Polish Notation''': The expression $(x-2)(x-1)$ would be written instead $\times \, - \, x \, 2 \, - \, x \, 1$, or something like that. There's no need for brackets or operator precedence, but it is hard to see at a glance what each operator applies to.

'''Add brackets''': around everything??

'''Make up a new rule''': At MathsJam Gathering 2020, Christian Lawson-Perfect suggested adding a rule "M before D except after 3". So $6 \div 2 \times 3 = 1$, but $3 \div 2 \times 4 = 6$.

==References==

Christian Lawson-Perfect has made a tool called [https://www.checkmyworking.com/misc/samdob/ SAMDOB] which lets you make up your own mnemonic and see how an expression is evaluated.

Colin Beveridge dubbed viral "puzzles" related to misunderstanding the order of operations [https://www.flyingcoloursmaths.co.uk/new-years-resolution-genius-sic/ fake maths].

Adam Townsend and Matthew Scroggs proposed using [https://chalkdustmagazine.com/blog/medusa-new-bodmas/ MEDUSA] instead of BODMAS to reduce ambiguities caused by division being done before multiplication or subtraction being doing before addition.

Kyle D. Evans' book, [https://atlantic-books.co.uk/book/maths-tricks-to-blow-your-mind/ Math Tricks to Blow Your Mind], collects many examples of viral order of operations problems in Chapter 4.

[[Category:Ambiguities]]

Missing multiplication symbol

2021-09-14T09:18:23Z

Christian Lawson-Perfect:

[[Category:Ambiguities]]

It's common to omit a multiplication symbol:

$ab = a \times b$

But sometimes it's not as clear:

Does $ a(b+1) = a \times (b+1)$, or is $a$ a function?<ref>[https://twitter.com/christianp/status/798843905231888385 Tweet by Christian Lawson-Perfect]</ref>

When writing a division on one line, does an implied multiplication bind more tightly than an explicit one?<ref>[https://twitter.com/christianp/status/1320650593241866241 Twitter thread by Christian Lawson-Perfect]</ref>

Is $a/bc$ equivalent to $\frac{a}{bc}$ or $\frac{a}{b}c$?

There seems to be an unwritten rule "juxtaposition is stickier". (See [[Juxtaposition means combine in the obvious way]])

But that might not apply when there are numbers involved: almost everyone would interpret $2/3x$ as $\frac{2}{3}x$ instead of $\frac{2}{3x}$

The "juxtaposition is stickier" rule only seems to break ties, not override the normal [[order of operations]]:

\[ ab^2 = a \times (b^2) \]

Sometimes the ambiguity comes from mistaking a function for an operation:

\[ (a+b) \Phi (a+b)\]

which can be viewed as either $ (a+b)\cdot \Phi(a+b)$, or $\Phi$ as binary addition-like operation, similar to $ (a+b)\oplus (a+b)$.<ref>Igor Pak, [https://scholarship.claremont.edu/jhm/vol8/iss1/14/ How to Write a Clear Math Paper: Some 21st Century Tips]</ref>

Jim Simons reckons we should give up on implicit multiplicatoin altogether.<ref>The Times, They Are A-Changin', Jim Simons, Mathematics in School, November 2020.</ref>

==References==
<references/>

Space is significant

2021-09-13T10:27:23Z

Christian Lawson-Perfect:

[[Category:Ambiguities]]

Empty space is often significant in mathematical notation, but it's easy to misinterpret, and hard to produce accurately.

TeX applies different amounts of spacing around symbols based on context. The ubiquity of TeX means that typeset expressions without this spacing look odd.

There's some evidence that people learning algebra use the spacing of symbols to deduce rules.<ref>[https://www.jstor.org/stable/30034809 Visual salience of Algebraic Transformations], David Kirshner and Thomas Awtry, Journal for Research in Mathematics Education.</ref>

==Examples==

Entries in a matrix are separated by empty space:

\[ \begin{pmatrix} 1 & - 2 \\ 3 & - 4 \end{pmatrix} \]

\[ \begin{pmatrix} 1-2 \\ 3 - 4 \end{pmatrix} \]

<hr/>

Statements about a line are usually separated from the main expression by a large space.

\[ 4=1 \mod 3 \]

versus

\[ 4=1 \bmod 3\]

<hr/>

Juxtaposing two fractions:

\[ \frac{1}{2} \frac{-3}{4} \]

\[ \frac{1-3}{2\, 4} \]

==References==

<references/>

The order of terms matters even when they commute

2021-09-06T13:01:25Z

Christian Lawson-Perfect:

[[Category:Inconsistencies]]
[[Category:Unspoken conventions]]

When writing an expression that consists of several terms, the conventions regarding their order appear arbitrary.

==Multiplication==

It is usual to write:

* $xy$ and $yx$ in either order;
* $5t$ but not $t5$ (to avoid confusion with $t_5$ or $t^5$);
* $x\sqrt{2}$ but not $\sqrt{2}x$ (to avoid confusion with $\sqrt{2x}$). (See [[Something on the right of a radical]])
* $\sqrt{2}\sin x$ but not $\sin x \sqrt{2}$ (to avoid confusion with $\sin \left(x\sqrt{2}\right)$). (See [[Function application without parentheses]])

==Addition==

People sometimes rearrange a sum to avoid a leading unary minus, even when this contradicts the convention of writing terms in decreasing order of degree:

\[ 1 - x \]

instead of

\[ -x + 1 \]

==Polynomials==

When writing a polynomial on its own, the usual convention is to write the terms in decreasing order of degree:

\[ x^3 - 32x^2 + 3x -1 \]

But for a series expansion, where higher-order powers are often omitted, it makes more sense to start with the lowest-degree term:

\[ -1 + 3x - 32x^2 + x^3 \]

Equation

2021-09-02T12:29:41Z

Christian Lawson-Perfect:

What is, and what isn't, an equation? <ref>[[https://twitter.com/pwr2dppl/status/1433190144404766721?t=i-tv-gxBN0q-FyoTknfGaA tweet by Piper H]]</ref>

[[Category:Ambiguities]]
[[Category:Conflicting definitions]]

Equation

2021-09-02T05:35:15Z

Christian Lawson-Perfect: Created page with "What is, and what isn't, an equation? <ref>https://twitter.com/pwr2dppl/status/1433190144404766721?t=i-tv-gxBN0q-FyoTknfGaA tweet by Piper H</ref>"

What is, and what isn't, an equation? <ref>[[https://twitter.com/pwr2dppl/status/1433190144404766721?t=i-tv-gxBN0q-FyoTknfGaA tweet by Piper H]]</ref>

Natural numbers

2021-08-26T13:57:53Z

Christian Lawson-Perfect:

The set of natural numbers, usually written $\mathbb{N}$, sometimes includes 0 and sometimes doesn't include 0.

Relatedly, the set of integers $\mathbb{Z}$ contains a subset $\mathbb{Z}^+$. Sometimes this is used to mean strictly positive integers and so doesn't contain 0 and sometimes it is used to mean non-negative integers and does contain 0.

Sometimes authors define $\mathbb{N}$ and $\mathbb{Z}^+$ as the same thing, either with or without zero<ref>[https://twitter.com/Joel_Feinstein/status/1430881362727288834 Tweet by Joel Feinstein]</ref>. Sometimes authors define them differently such that $\mathbb{N}$ contains zero and $\mathbb{Z}^+$ does not, and sometimes authors define them differently such that $\mathbb{N}$ does not contain zero and $\mathbb{Z}^+$ does.

[[Category:Conflicting definitions]]

Unary division

2021-08-11T04:47:00Z

Christian Lawson-Perfect:

A minus sign with nothing on the left represents negation of whatever's on the right.

Why isn't there a unary division symbol?<ref>[https://twitter.com/christianp/status/478815299081633793 Tweet by Christian Lawson-Perfect]: "We have unary minus, i.e. "-2" is the same as "0-2". Why don't we have unary division, i.e. "÷2" could mean the same as "1÷2"?"</ref><ref>[https://twitter.com/christianp/status/1061949551949557760 Tweet]: "Years after impishly proposing a 'unary division' operator at big #mathsjam, I find myself actually needing to introduce one in my real work."</ref>

\[ \div x = \frac{1}{x} \]

The page on wheels<ref>https://ncatlab.org/nlab/show/wheel</ref> in the nLab suggests exactly this.

[[Category:Inconsistencies]]

Bang

2021-07-29T15:12:37Z

Christian Lawson-Perfect: Redirected page to !

#REDIRECT [[!]]

!

2021-07-29T15:11:09Z

Christian Lawson-Perfect:

[[Category:Ambiguities]]

The ! symbol is used to represent the factorial operation.

When a factorial appears inside a sentence, it's possible to misinterpret the ! as an exclamation mark<ref>[https://twitter.com/matthras/status/1415236669553274882 Tweet by Matt Mack]</ref>:

<blockquote>
"How many ways of ordering six objects are there?"

"There are 6!"
</blockquote>

Two ! symbols together represent the ''double factorial'', multiplying just the odd or even numbers. So juxtaposition doesn't represent composition here: $x!! \neq (x!)!$

A ! symbol on the left represents the number of derangements, or ''subfactorial''. The order of precedence is not clear:

Does $!n!\ = (!n)!$ or $!(n!)$?

Does $a!b = (a!)b $ or $a(!b)$?

Does it make it clearer that a factorial is a present if you add another punctuation symbol after the ! symbol?

<blockquote>
"There are 6!."
</blockquote>

However, if you want to express surprise with an exclamation mark, it could look like a double factorial:

<blockquote>
"There are 6!!"
</blockquote>

Maybe ! should only be used for "factorial" in contexts that are unambiguously and clearly delimited mathematical notation, and the word "factorial" should be used in prose:

<blockquote>
"There are 6 factorial."
</blockquote>

Easily confused symbols

2021-07-29T15:06:02Z

Christian Lawson-Perfect:

[[Category:Handwriting]]
[[Category:Needs filling in]]

Some symbols are hard to distinguish, or are easily confused for each other, particularly when handwritten.

==Handwritten examples==

When 1 is written with a twiddle at the top rather than just as a straight line, it could look like a 7. Or, a hastily-written 7 could look like this kind of 1.

A common way of resolving this is to draw a line across the middle of the 7.

<hr/>

m, μ, u and n can all run into each other, particularly when written in cursive.

A famous example where these combine to produce something hard to read is the word 'minimum':

[[File:Minimum-VPantaloni-handwriting.png|thumb|The word "minimum", handwritten.]]

Easily confused symbols

2021-07-29T14:59:41Z

Christian Lawson-Perfect: Created page with "Category:Handwriting Some symbols are hard to distinguish, or are easily confused for each other, particularly when handwritten. ==Handwritten examples== When 1 is writ..."

[[Category:Handwriting]]

Some symbols are hard to distinguish, or are easily confused for each other, particularly when handwritten.

==Handwritten examples==

When 1 is written with a twiddle at the top rather than just as a straight line, it could look like a 7. Or, a hastily-written 7 could look like this kind of 1.

A common way of resolving this is to draw a line across the middle of the 7.

<hr/>

m, μ, u and n can all run into each other, particularly when written in cursive.

A famous example where these combine to produce something hard to read is the word 'minimum':

[[File:Minimum-VPantaloni-handwriting.png|thumb|The word "minimum", handwritten.]]

Category:Handwriting

2021-07-29T14:54:23Z

Christian Lawson-Perfect:

This category collects things that only or mainly come up in handwritten notation.

Juxtaposition means combine in the obvious way

2021-07-16T10:17:06Z

Christian Lawson-Perfect:

Missing multiplication symbol

2021-07-16T10:16:37Z

Christian Lawson-Perfect:

[[Category:Ambiguities]]

It's common to omit a multiplication symbol:

$ab = a \times b$

But sometimes it's not as clear:

Does $ a(b+1) = a \times (b+1)$, or is $a$ a function?<ref>[https://twitter.com/christianp/status/798843905231888385 Tweet by Christian Lawson-Perfect]</ref>

When writing a division on one line, does an implied multiplication bind more tightly than an explicit one?<ref>[https://twitter.com/christianp/status/1320650593241866241 Twitter thread by Christian Lawson-Perfect]</ref>

Is $a/bc$ equivalent to $\frac{a}{bc}$ or $\frac{a}{b}c$?

There seems to be an unwritten rule "juxtaposition is stickier". (See [[Juxtaposition means combine in the obvious way]])

But that might not apply when there are numbers involved: almost everyone would interpret $2/3x$ as $\frac{2}{3}x$ instead of $\frac{2}{3x}$

The "juxtaposition is stickier" rule only seems to break ties, not override the normal [[order of operations]]:

\[ ab^2 = a \times (b^2) \]

Sometimes the ambiguity comes from mistaking a function for an operation:

\[ (a+b) \Phi (a+b)\]

which can be viewed as either $ (a+b)\cdot \Phi(a+b)$, or $\Phi$ as binary addition-like operation, similar to $ (a+b)\oplus (a+b)$.<ref>Igor Pak, [https://scholarship.claremont.edu/jhm/vol8/iss1/14/ How to Write a Clear Math Paper: Some 21st Century Tips]</ref>
==References==
<references/>

The order of terms matters even when they commute

2021-07-16T09:02:43Z

Christian Lawson-Perfect:

[[Category:Inconsistencies]]
[[Category:Unspoken conventions]]

When writing an expression that consists of several terms, the conventions regarding their order appear arbitrary.

==Multiplication==

It is usual to write:

* $xy$ and $yx$ in either order;
* $5t$ but not $t5$ (to avoid confusion with $t_5$ or $t^5$);
* $x\sqrt{2}$ but not $\sqrt{2}x$ (to avoid confusion with $\sqrt{2x}$). (See [[Something on the right of a radical]])
* $\sqrt{2}\sin x$ but not $\sin x \sqrt{2}$ (to avoid confusion with $\sin \left(x\sqrt{2}\right)$).

==Addition==

People sometimes rearrange a sum to avoid a leading unary minus, even when this contradicts the convention of writing terms in decreasing order of degree:

\[ 1 - x \]

instead of

\[ -x + 1 \]

==Polynomials==

When writing a polynomial on its own, the usual convention is to write the terms in decreasing order of degree:

\[ x^3 - 32x^2 + 3x -1 \]

But for a series expansion, where higher-order powers are often omitted, it makes more sense to start with the lowest-degree term:

\[ -1 + 3x - 32x^2 + x^3 \]