Editing 2319: Large Number Formats

This comic shows what the way you write large numbers says about you. Different people use different methods to express large numbers. And this comic claims it can tell something about you based on the way you format large numbers. In this way, the comic is similar in idea to [[977: Map Projections]], where it was your choice of map projections that could tell something about you.

| This is the full number, 25259974097204, 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 anglophone world; conventions for writing large numbers in full vary considerably across cultures. For example, in countries where the period 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 2,52,59,97,40,97,204 or “two nil, fifty-two kharab, fifty-nine arab, ninety-seven crore, forty lakh, ninety-seven thousand, two hundred and four.”

| In current English usage, across the Anglophonic world with some hold-outs, an n-illion means 10^(3n+3) as per the {{w|short scale}} system popularised by American influence in international trade, so a trillion means 10^12, as above. However, older British English use had an n-illion meaning 10^(6n) (i.e. the simpler calculation of ''million^n''), so a billion meant 10^12. The change stems from a 1974 commitment by Harold Wilson, the Prime Minister of the UK at the time, to change from the {{w|long scale}} (previously often described as the British system) to the short one for all official purposes.

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.

|2.526×10¹³

| 2.525997×10¹³

| 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 ''2.5997x10¹³''), but Randall updated the comic.

| 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 most common programming languages]. 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 could 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 (although in many situations  the compiler or preprocessor would detect this and solve it correctly, making it functionally identical to the first case). A software developer might write it that way if they are a novice who is not familiar with the first notation, or they could simply have an personal preference that considers the second version easier to read. Perhaps an additional 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.  

| 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^''e'', 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 &times; 2⁴⁵ and 0.717931688 &times; 2⁴⁵; 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 {{w|hexadecimal}}, are <tt>0.b7ca5e</tt> &times; 2^<tt></tt>2d and <tt>0.b7ca5f</tt> &times; 2^<tt></tt>2d.)

 

| For extremely large distances, astronomers typically only care about orders of magnitude, e.g. whether a number is 10¹³, as opposed to 10¹² or 10¹⁴. Randall often jokes about the lack of precision needed by astronomers, such as in [[2205: Types of Approximation]] 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.

| The natural numbers can be constructed in a {{w|set theory}} in various ways. In the most common of these, the {{w|Natural_number#Von_Neumann_ordinals|Von Neumann ordinals}}, 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 a number in this form requires an exponential number of characters - that is, ''n'' + 1 requires over twice the characters as ''n'' does to write out. Thus, this method could not be finished, as it would require more data to be stored than there is matter in the universe to store it.

| 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 this case 1,262,998,704,860 * 20 + 4 yields the exact number.

| 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''^{''e''^''e''⁷⁹}, but is more commonly approximated as 10^{10^10³⁴}. 13.4024 is a rounded version of the {{w|common logarithm}} of 25,259,974,097,204 (log₁₀ 25,259,974,097,204 = 13.4024329009); thus, this "format" is still mathematically correct, but uncommon. However, only by using many more digits will the result get close enough to be rounded to the original number 10^13.40243290087302 = 25,259,974,097,203.5, which would round up to the correct number. The number from the title text, 10^13.4024 = 25,258,060,548,319.6, differs from the original number by over a billion.

:[A panel only with text. At the top there is four lines of explanatory text. Below that are 2 columns with 5 rows of number formats. 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>

:(Using the approximate current distance to Jupiter in inches as an example)

:<span style="color:#ba0000">25,259,974,097,204</span>

 

:<span style="color:#ba0000">25 trillion</span>

:<span style="color:#ba0000">25 billion</span>

 

:<span style="color:#ba0000">2.526x10¹³</span>

:<span style="color:#ba0000">2.525997x10¹³</span>

 

:<span style="color:#ba0000">2.526e13 or 2.526*10^13</span>

:<span style="color:#ba0000">25,259,973,541,888</span>

 

:<span style="color:#ba0000">10¹³</span>

:<span style="color:#ba0000">{∅,{∅},{∅,{∅}},{∅,{∅},{...</span>

 

:<span style="color:#ba0000">1,262,998,704,860 score and four</span>

[[Category:Programming]]

[[Category:Math]]

[[Category:Astronomy]]

[[Category:Science]]