Editing 2687: Division Notation
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==Explanation== | ==Explanation== | ||
− | + | {{incomplete|Created by a GROUP OF SCHOOLCHILDREN DIVIDED AMONGST THEMSELVES. Do NOT delete this tag too soon.}} | |
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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] | 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] | ||
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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 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 | + | The expression on the fourth line, <sup>A</sup>/<sub>B</sub>, is how division is usually written by hand. 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. It takes more effort to type with this notation; however, the Unicode character sets provide some specific fractions such as ⅓ as well as some superscript and subscript characters, so someone familiar with it might use it to write fractions such as ²²⁄₇. |
− | The fifth notation is the way division is written in science | + | The fifth notation is the way division is written in science: <table style="display: inline-table; line-height: 0.6em; vertical-align: middle; font-size:7pt; 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 seperating 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 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 | + | The AB<sup>-1</sup> format is also often used to express physical units. |
− | The | + | 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). Randall warns the reader they should escape while they still can, because both the function itself and the math environment as a whole are going to get relatively tedious. Integer division can be defined in terms of multiplicative inequalities and the remainder, or modulo ('%' in Python), 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. |
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+ | The title text 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 percent sign. A scientist might be really mad at the use of the division sign as an alternative to a percent sign not just because it is the wrong symbol in that context, but also because of the tedium of modern consumerist practices. | ||
==Transcript== | ==Transcript== | ||
− | + | {{incomplete transcript|Do NOT delete this tag too soon.}} | |
− | : | + | :<u>Division notation</u> |
− | + | :A÷B | |
− | + | :B⟌A Schoolchild. | |
− | : | + | :A/B Software engineer. |
− | + | <span style="display:inline-flex;flex-direction:column;vertical-align:middle;"> | |
− | :A/B | + | <span>A</span> |
− | + | <span style="border-top:1px solid">B</span> | |
− | + | </span> Scientist | |
− | + | :AB<sup>-1</sup> Fancy scientist. | |
− | + | :F(A, B) such that F(G)= (text getting smaller) Oh no, run | |
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− | :F(A, B) | ||
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{{comic discussion}} | {{comic discussion}} |