Editing 2687: Division Notation

{{comic
| number    = 2687
| date      = October 19, 2022
| title     = Division Notation
| image     = division_notation_new_2x.png
| imagesize = 235x310px
| noexpand  = true
| titletext = Science tip: Scientists hardly ever use the two-dot division sign, and when they do it often doesn't even mean division, but they still get REALLY mad when you repurpose it to write stuff like SALE! ALL SHOES 30÷ OFF!
}}

==Explanation==

In this comic there is another one of [[Randall|Randall's]] [[:Category:Tips|Tips]], this time a [[:Category:Science tip|Science Tip]], however, it is only mentioned in the title text. See below for more.

This comic pokes fun at some of the ways to write the {{w|Division (mathematics)|division}} operation in math. In this comic, [[Randall]] has used A as the dividend (the number being divided) and B as the divisor (the number that A is divided by). Division is the fourth simplest arithmetic operation in mathematics, after addition, subtraction, and multiplication.[https://plato.stanford.edu/entries/principia-mathematica/#PartIVRelaArit]

The first two of the seven notations shown are the {{w|division sign}} (÷) and the {{w|radix}} used for {{w|short division}} and {{w|long division}} in beginning arithmetic. (Note: the short/long division radix is only used in some countries, and there are [https://en.wikipedia.org/wiki/Long_division#Notation_in_non-English-speaking_countries different notations in the non-English speaking world]). These methods of division are often used by school children because the ÷ sign is what most people use when first learning division, and the short division radix is usually the first algorithm learned for dividing arbitrary dividends, typically starting with the easier abbreviated short division form.

The expression on the third line, A/B, is the way division is usually written in software code. The four simple arithmetic operations in programming usually are +, -, *, /. This line was not in the [[media:division_notation_2x.png|first version]] of the comic. This is most commonly seen in regular mathematics as it somewhat saves space, and is easy to type with the slash key. Additionally, it uses standard {{w|ASCII}} characters instead of sophisticated notation. A notable exception is {{w|APL_(programming_language)|APL}}, which uses an idiosyncratic character set modeled after traditional arithmetic.

The expression on the fourth line, <sup>A</sup>/<sub>B</sub>, is how division is usually written by hand{{Actual citation needed}}. It is nearly identical to the fraction notation that follows, but the diagonal line allows each number to be bigger while still fitting into a single line of text. The third line's representation is the best approximation of this line's notation on a computer, without using more obscure Unicode characters. The Unicode character set can be used to accurately represent division as on the fourth line in plain text, using a small set of precomposed fractions (⅔, ⅕, etc.), regular numerals and [https://unicode-explorer.com/c/2044 U+2044 FRACTION SLASH] (e.g. 22⁄7, provided font support exists), or superscript and subscript numerals (e.g. ²²/₇, or ²²⁄₇ with the fraction slash). Using any of these requires a greater knowledge of Unicode, and the know-how (and, possibly, patience) to type them, so it is likely that only a Unicode enthusiast would type division like this, rather than an alternate or equivalent format such as <sup>22</sup>/<sub>7</sub> as rendered in HTML.

The fifth notation is the way division is written in science and intermediate and higher mathematics: <table style="display: inline-table; line-height: 0.75em; vertical-align: middle; font-size:10pt; text-size-adjust: none;"><tr><td><u>A</u></td></tr><tr><td>B</td></tr></table>, i.e. the dividend on the top of the expression, over the divisor on the bottom with a horizontal line separating them. This is how a {{w|Fraction|fraction}} would be written. It has the advantage of clearly separating the numerator and denominator when they are longer expressions, such as polynomials, without needing to add parentheses. This format is mostly used in written and professionally typeset math, as it can't be typed without something like {{w|MathML}}, {{w|LaTeX}} or HTML tables.

The sixth, "fancy," notation uses a negative exponent. The exponent -1 is equivalent to {{w|Multiplicative inverse|reciprocation}}. It can be used to keep an entire division expression on one line. Note that AB<sup>-1</sup> is equal to <sup>A</sup>/<sub>B</sub> only if A and B are in a commutative ring (and B has an inverse). If A and B are, for example, matrices, AB<sup>-1</sup> is not the same in general as B<sup>-1</sup>A (and the notation <sup>A</sup>/<sub>B</sub> is never used in this case as it would be ambiguous). The AB<sup>-1</sup> format is also often used to express physical units.

The final form of notation declares a function. The writer defines a new function, F, that takes in the parameters A and B, before listing out the function's definition (trailing off in increasingly smaller text). Defining things as functions is very normal for mathematics, but likely not very efficient at all to commonly do for the simple operation of division.  Occasionally a highly formalized definition such as this could be a setup for an elaborate and extensive proof, hence Randall warns the reader they should escape while they still can. Integer division can be defined in terms of multiplicative inequalities and the remainder, or modulo ('%' in most programming languages) operator. This situation is likely to occur in many sorts of algebra, where one might have to define what "division" means for two elements of a mathematical object such as a group, ring, or magma. One example would be an object G, such that, for two elements A and B of G, "A divided by B" is defined as an element C such that CB=A, or alternatively as an element C such that BC=A. These definitions will differ if multiplication in G is not commutative. Furthermore, if such a C is not unique, the function F(A,B) will need to include a method to select a unique value for "A divided by B" for each A and B. Thus, the F(A,B) in the comic might not even refer to a uniquely defined operation, but simply to the property of a function F(A,B) that is a valid division operation on G, given some definition of division. You were warned.

The title text is a [[:Category:Science tip|Science Tip]]. It discusses how the division sign (÷) has fallen out of favor in most professional contexts (the ISO-80000 guidelines even specify the symbol "should not be used") yet has resisted all efforts to repurpose it as a new function. Specifically, it pokes fun at how similar the division sign is to a {{w|percent sign}} (%). A scientist might be really mad at the use of the division sign as an alternative to a percent sign because it is the wrong symbol in that context, ever though they never use it for the original meaning any more. The use of ÷ instead of % is something that may appear on signs for discount offers.

==Transcript==
:[The transcript has both formatted text for visuals and a description at the start of the line.]
:[Underlined text:] <u>Division notation</u> 

:[A [Division sign] B is above, B [Long division symbol] A is below. They are connected by a close brace.] <table style="display: inline-table; line-height: 1em; vertical-align: middle; font-size:10pt; text-size-adjust: none;"><tr><td>A÷B</td></tr><tr><td>B⟌A</td></tr></table>} 
:[Label on the right:] Schoolchild

:A/B 
:[Label:] Software engineer

:[A/B except the A is slightly above and the B is slightly below, and the slash is more diagonal to fit them.] <sup>A</sup>⁄<sub>B</sub> 
:[Label:] Normal person or Unicode enthusiast

:[A fraction with A on the top and B on the bottom.] <table style="display: inline-table; line-height: 0.8em; vertical-align: middle; font-size:10pt; text-size-adjust: none;"><tr><td><u>A</u></td></tr><tr><td>B</td></tr></table> 
:[Label:] Scientist

:[AB with -1 in superscript.] AB<sup>-1</sup> 
:[Label:] Fancy scientist

:F(A, B) [text gets smaller] such that [text gets smaller] F(G)= [Text is too small to read] 
:[Label:] Oh no, run

{{comic discussion}}

[[Category:Math]]
[[Category:Science]]
[[Category:Programming]]
[[Category:Science tip]]