Editing 2319: Large Number Formats
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==Explanation== | ==Explanation== | ||
− | This comic shows | + | {{incomplete|Created by ABRAHAM LINCOLN. Please mention here why this explanation isn't complete. Do NOT delete this tag too soon.}} |
+ | This comic shows how different people express large numbers. This number in question is approximately the distance from the planet Earth to the planet Jupiter as of June 2020, in {{w|inch|inches}} (1 inch = 2.54 cm). | ||
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{| class="wikitable" | {| class="wikitable" | ||
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| 25,259,974,097,204 | | 25,259,974,097,204 | ||
| Normal Person | | Normal Person | ||
− | | This is the full number | + | | This is the full number, written out in the normal fashion, with commas to indicate powers of 1000. Although writing out the number in full is indeed a common action for normal people, the specific comma convention depicted here is only considered normal in the Anglo-Saxon world; conventions for writing large numbers in full vary considerably across cultures. For example, in countries where the comma is used as a {{w|decimal separator}} (including Europe outside the UK), one would write the number as 25.259.974.097.204 (or 25'259'974'097'204 in Switzerland, or 25 259 974 097 204 in Poland, France and Estonia). Under the {{w|Indian numbering system}}, this number would be written as 25,25,997,40,97,204. |
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| 25 Trillion | | 25 Trillion | ||
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As well as 'traditionalist' British use, the long scale is widely used in the non-Anglophone world, in local language versions, though while the British system tended to infill n-and-a-half powers of the million with the term "thousand n-illion", the suffix "-illi''ard''", or equivalent, is often used for the thousands multiple directly atop the respective "-illion" point. | As well as 'traditionalist' British use, the long scale is widely used in the non-Anglophone world, in local language versions, though while the British system tended to infill n-and-a-half powers of the million with the term "thousand n-illion", the suffix "-illi''ard''", or equivalent, is often used for the thousands multiple directly atop the respective "-illion" point. | ||
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− | |2. | + | |2.526x10<sup>13</sup> |
|Scientist | |Scientist | ||
|This number is formatted in {{w|scientific notation}}, using the exponent 10<sup>13</sup>. | |This number is formatted in {{w|scientific notation}}, using the exponent 10<sup>13</sup>. | ||
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− | | 2. | + | | 2.525997x10<sup>13</sup> |
| Scientist trying to avoid rounding up | | Scientist trying to avoid rounding up | ||
| Using as many decimal places as necessary until hitting a digit (0-4) that results in rounding down, even if it goes against the common scientific practice of reporting the correct amount of "significant figures". [[:File:large number formats.png|A previous version of the comic]] had a typo (the number was ''2.5997x10<sup>13</sup>''), but Randall updated the comic. | | Using as many decimal places as necessary until hitting a digit (0-4) that results in rounding down, even if it goes against the common scientific practice of reporting the correct amount of "significant figures". [[:File:large number formats.png|A previous version of the comic]] had a typo (the number was ''2.5997x10<sup>13</sup>''), but Randall updated the comic. | ||
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2.526*10^13 | 2.526*10^13 | ||
| Software developer | | Software developer | ||
− | | The first example is how the number would be expressed as a floating point number in scientific notation in [https://rosettacode.org/wiki/Literals/Floating_point | + | | The first example is how the number would be expressed as a floating point number in scientific notation in [https://rosettacode.org/wiki/Literals/Floating_point every common programming language]. The second example is a technically correct way of expressing the same thing in some programming languages in which exponentiation is indicated by the ^ operator. However writing it that way instead of the first way would be considered quirky, as it is written as an instruction to the computer to calculate the product of a number with 10 raised to power 13, instead of just writing the number. A software developer might write it that way if they are a novice who is not familiar with the first notation. Or they could have an unusual personal preference that considers the second version easier to read. Perhaps the joke for the second version is that it is the standard scientific notation with the x for multiplication and superscript for raising to a power replaced with the notation used in many programming languages of * and ^, i.e., a software developer writing down a number in scientific notation, not necessarily while writing a program, would by habit write a * for multiplication and a ^ for exponentiation. |
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| 25,259,973,541,888 | | 25,259,973,541,888 | ||
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| The two most common computer {{w|Floating-point arithmetic|floating-point}} formats are the IEEE 754 {{w|Single-precision floating-point format|single-precision}} and {{w|Double-precision floating-point format|double-precision}} representations. These are ''binary'' floating-point formats, representing numbers as the quantity ''a'' × 2<sup>''e''</sup>, for some fractional number ''a'' and exponent ''e''. Both the values ''a'' and ''e'' have a fixed size in bits, and therefore a finite range. In single-precision, ''a'' and ''e'' have (effectively) 24 and 8 bits, respectively, while in double precision the effective sizes are 53 and 11 bits. | | The two most common computer {{w|Floating-point arithmetic|floating-point}} formats are the IEEE 754 {{w|Single-precision floating-point format|single-precision}} and {{w|Double-precision floating-point format|double-precision}} representations. These are ''binary'' floating-point formats, representing numbers as the quantity ''a'' × 2<sup>''e''</sup>, for some fractional number ''a'' and exponent ''e''. Both the values ''a'' and ''e'' have a fixed size in bits, and therefore a finite range. In single-precision, ''a'' and ''e'' have (effectively) 24 and 8 bits, respectively, while in double precision the effective sizes are 53 and 11 bits. | ||
− | Fully representing the number 25,259,974,097,204 (in any format) requires at least 45 bits. Therefore this number cannot be represented exactly as a single-precision float. The closest possible representations are 0.717931628 × 2<sup>45</sup> and 0.717931688 × 2<sup>45</sup>; these work out to 25,259,973,541,888 and 25,259,975,639,040, respectively. Of these, the one ending in 888 is considerably closer to the original, so is chosen due to {{w|rounding}}. (Naturally these numbers are represented internally in binary, not decimal; the actual representations, in | + | Fully representing the number 25,259,974,097,204 (in any format) requires at least 45 bits. Therefore this number cannot be represented exactly as a single-precision float. The closest possible representations are 0.717931628 × 2<sup>45</sup> and 0.717931688 × 2<sup>45</sup>; these work out to 25,259,973,541,888 and 25,259,975,639,040, respectively. Of these, the one ending in 888 is considerably closer to the original, so is chosen due to {{w|rounding}}. (Naturally these numbers are represented internally in binary, not decimal; the actual representations, in hexadecimal, are <tt>0x0.b7ca5e</tt> × 2<sup><tt></tt>0x2d</sup> and <tt>0x0.b7ca5f</tt> × 2<sup><tt></tt>0x2d</sup>.) |
− | In many programming languages, the keyword to request a single-precision floating-point variable is <tt>float</tt>, while the keyword to request double-precision is <tt>double</tt>. It is an easy mistake to make to forget about the limited precision available with type <tt>float</tt>, especially since its name sounds like what you want for "floating point". (Had the programmer remembered to use type <tt>double</tt>, the number 25,259,974,097,204 could have been represented exactly | + | In many programming languages, the keyword to request a single-precision floating-point variable is <tt>float</tt>, while the keyword to request double-precision is <tt>double</tt>. It is an easy mistake to make to forget about the limited precision available with type <tt>float</tt>, especially since its name sounds like what you want for "floating point". (Had the programmer remembered to use type <tt>double</tt>, the number 25,259,974,097,204 could have been represented exactly, as <tt>0x0.b7ca5e43c9a000</tt> × 2<sup><tt></tt>0x2d</sup>.) |
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| 10<sup>13</sup> | | 10<sup>13</sup> | ||
| Astronomer | | Astronomer | ||
− | | For extremely large distances, astronomers typically only care about orders of magnitude, e.g. whether a number is 10<sup>13</sup>, as opposed to 10<sup>12</sup> or 10<sup>14</sup>. Randall often jokes about the lack of precision needed by astronomers, such as in [[2205 | + | | For extremely large distances, astronomers typically only care about orders of magnitude, e.g. whether a number is 10<sup>13</sup>, as opposed to 10<sup>12</sup> or 10<sup>14</sup>. Randall often jokes about the lack of precision needed by astronomers, such as in xkcd #[[2205]] where the astronomer-cosmologist is equally willing to make pi equal to one, or ten. The original number is rounded to the nearest power of ten. |
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| {∅,{∅},{∅,{∅}},{∅,{∅},{... | | {∅,{∅},{∅,{∅}},{∅,{∅},{... | ||
| Set theorist | | Set theorist | ||
− | | | + | | In {{w|Zermelo–Fraenkel set theory}}, the natural numbers are defined recursively by letting 0 = ∅ (the {{w|empty set}}), and ''n'' + 1 = ''n'' ∪ {''n''}. So, every natural number ''n'' is the set of all natural numbers less than ''n'', and since 0 is defined as the empty set, all numbers are nested sets of empty sets. Note that writing out the full number in this fashion would take more than its square in number of characters; that is to say, if each character took up one square centimeter, this "number" would not fit on a square piece of paper whose edge reached to Jupiter. |
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| 1,262,998,704,860 score and four | | 1,262,998,704,860 score and four | ||
| Abraham Lincoln | | Abraham Lincoln | ||
− | | In the {{W|Gettysburg Address}}, Lincoln speaks the number "87" as "four score and seven" ("score" meaning "20"). Base-20 or {{w|vigesimal}} numeral systems are or have been used in pre-Columbian-American, African and many other cultures. In French it is used only for higher numbers (e.g. 92 = quatre-vingt-douze). In English it can appear in certain archaic and classic contexts, such as the King James translation of the Bible ("threescore years and ten" to be the life expectancy of a human according to Psalm 90:10). In these cases, a number is written in "score" (multiples of 20) plus a remainder. | + | | In the {{W|Gettysburg Address}}, Lincoln speaks the number "87" as "four score and seven" ("score" meaning "20"). Base-20 or {{w|vigesimal}} numeral systems are or have been used in pre-Columbian-American, African and many other cultures. In French it is used only for higher numbers (e.g. 92 = quatre-vingt-douze). In English it can appear in certain archaic and classic contexts, such as the King James translation of the Bible ("threescore years and ten" to be the life expectancy of a human according to Psalm 90:10). In these cases, a number is written in "score" (multiples of 20) plus a remainder. |
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| 10^13.4024 ''(title text)'' | | 10^13.4024 ''(title text)'' | ||
| A person who has come back to numbers after a journey deep into some random theoretical field | | A person who has come back to numbers after a journey deep into some random theoretical field | ||
− | | In some fields of mathematics, especially those dealing with very {{w|large numbers}}, numbers are sometimes represented by raising ten (or some other convenient base) to an oddly precise power, to facilitate comparison of their magnitudes without filling up pages upon pages of digits. An example of this is {{w|Skewes's number}}, which is formally calculated to be ''e''<sup>''e''<sup>''e''<sup>79</sup></sup></sup>, but is more commonly approximated as 10<sup>10<sup>10<sup>34</sup></sup></sup>. 13.4024 is | + | | In some fields of mathematics, especially those dealing with very {{w|large numbers}}, numbers are sometimes represented by raising ten (or some other convenient base) to an oddly precise power, to facilitate comparison of their magnitudes without filling up pages upon pages of digits. An example of this is {{w|Skewes's number}}, which is formally calculated to be ''e''<sup>''e''<sup>''e''<sup>79</sup></sup></sup>, but is more commonly approximated as 10<sup>10<sup>10<sup>34</sup></sup></sup>. 13.4024 is the {{w|common logarithm}} of 25,259,974,097,204 (log<sub>10</sub> 25,259,974,097,204 = 13.4024329009); thus, this "format" is still mathematically correct, but uncommon. |
|} | |} | ||
==Transcript== | ==Transcript== | ||
− | :[A panel only with text. At the top there is four lines of explanatory text. Below that are | + | {{incomplete transcript|Do NOT delete this tag too soon.}} |
+ | :[A panel only with text. At the top there is four lines of explanatory text. Below that there are 5 rows of number formats. There are 2 columns in each row. Each numerical format is in red, with black text explaining the format below it.] | ||
− | :<big>What the way you write large numbers says about you</big> | + | :<big>What the way you write large</big> |
− | :(Using the approximate current distance to Jupiter in inches as an example) | + | :<big>numbers says about you</big> |
+ | :(Using the approximate current distance | ||
+ | :to Jupiter in inches as an example) | ||
− | :<span style="color: | + | :[First row:] |
+ | :<span style="color:red">25,259,974,097,204</span> | ||
:Normal person | :Normal person | ||
− | + | :<span style="color:red">25 trillion</span> | |
− | :<span style="color: | ||
:Normal person | :Normal person | ||
− | :<span style="color: | + | :[Second row:] |
+ | :<span style="color:red">25 billion</span> | ||
:Old British person | :Old British person | ||
− | + | :<span style="color:red">2.526x10<sup>13</sup></span> | |
− | :<span style="color: | ||
:Scientist | :Scientist | ||
− | :<span style="color: | + | :[Third row:] |
+ | :<span style="color:red">2.525997x10<sup>13</sup></span> | ||
:Scientist trying to avoid rounding up | :Scientist trying to avoid rounding up | ||
− | + | :<span style="color:red">2.526e13 or<br>2.526*10^13</span> | |
− | :<span style="color: | ||
:Software developer | :Software developer | ||
− | :<span style="color: | + | :[Fourth row:] |
+ | :<span style="color:red">25,259,973,541,888</span> | ||
:Software developer who forgot about floats | :Software developer who forgot about floats | ||
− | + | :<span style="color:red">10<sup>13</sup></span> | |
− | :<span style="color: | ||
:Astronomer | :Astronomer | ||
− | :<span style="color: | + | :[Fifth row:] |
+ | :<span style="color:red">{∅,{∅},{∅,{∅}},{∅,{∅},{...</span> | ||
:Set theorist | :Set theorist | ||
− | + | :<span style="color:red">1,262,998,704,860<br>score and four</span> | |
− | :<span style="color: | ||
:Abraham Lincoln | :Abraham Lincoln | ||
{{comic discussion}} | {{comic discussion}} | ||
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[[Category:Comics with color]] | [[Category:Comics with color]] | ||
− | [[Category:Programming]] | + | [[Category: Programming]] |
− | [[Category:Math]] | + | [[Category: Math]] |
− | [[Category:Astronomy]] | + | [[Category: Astronomy]] |
− | [[Category:Science]] | + | [[Category: Science]] |
[[Category:Comics featuring politicians]] | [[Category:Comics featuring politicians]] | ||
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