Editing 2637: Roman Numerals
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==Explanation== | ==Explanation== | ||
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− | The modern system of representing numbers is a decimal positional notation using | + | Roman numerals are the system of representing numbers used during the Roman Empire. The letters I, V, X, L, C, D, and M are used to represent numbers, with each letter representing a consistent value. Specifically, I represents 1, V represents 5, X represents 10, L represents 50, C represents 100, D represents 500, and M represents 1000. One way of stating the rules for combining Roman numerals next to each other are that a Roman numeral is added to a Roman numeral of equal or lesser value just to its right (e.g., II=1+1=2 because 1≥1, and VI=5+1=6 because 5≥1), and a Roman number is subtracted from a Roman numeral of greater value just to its right (e.g., IV=5-1=4 because 1<5, and IX=10-1=9 because 1<10). (Also, each place must be written separately, e.g., one cannot represent 49 via IL but instead must represent the tens place and ones place separately via XL IX—although the space would not be included in practice). |
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+ | The modern system of representing numbers is a decimal positional notation using Hindu–Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9), which are so-called because they were invented in India. However, because they were introduced to Europe by Arabic merchants, Westerners often call them Arabic or Hindu-Arabic numerals. Instead of concatenating several 1s, the single character 2 represents 1+1, 3 represents 1+1+1, etc… all the way to 9 representing 1+1+1+1+1+1+1+1+1. Integers larger than nine are represented as a sum of digits multiplied by different powers of ten. Each time a digit is moved one place to the left, the value that it represents is multiplied by ten (e.g., moving 3 to the left, starting in the ones place, changes the value that it represents from three to three tens to three hundreds to three thousands…). Positional notations require a character for the additive identity, 0, to fill in any gaps so that the digits to its left are positioned correctly. The string "4096" represents 4×10<sup>3</sup>+0×10<sup>2</sup>+9×10<sup>1</sup>+6×10<sup>0</sup>. | ||
Thus in Roman numerals a digit always has the same absolute value but may be treated as positive or negative depending on the digit after it, whereas for Hindu-Arabic numerals, a digit's value changes by a power of 10 depending on its absolute position and is never subtracted. | Thus in Roman numerals a digit always has the same absolute value but may be treated as positive or negative depending on the digit after it, whereas for Hindu-Arabic numerals, a digit's value changes by a power of 10 depending on its absolute position and is never subtracted. | ||
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<center>IV + V = IX</center> | <center>IV + V = IX</center> | ||
− | Translated | + | Translated properly into more familiar digits, these equations are: |
<center>1 + 1 = 2</center> | <center>1 + 1 = 2</center> | ||
<center>2 + 2 = 4</center> | <center>2 + 2 = 4</center> | ||
<center>4 + 5 = 9</center> | <center>4 + 5 = 9</center> | ||
− | But Randall/Cueball replaced each letter individually with its value in Hindu-Arabic numerals — ignoring the abovementioned rules for interpreting combined Roman numbers, instead using | + | But Randall/Cueball replaced each letter individually with its value in Hindu-Arabic numerals — ignoring the abovementioned rules for interpreting combined Roman numbers, instead using simple concatenation. Specifically, "I" is replaced with "1", "V" is replaced with "5", and "X" is replaced with "10". For example, for IX at the end of the last equation, "I" is replaced with "1", and "X" is replaced with "10", so "IX" becomes "110". Thus, the equations become |
<center>1 + 1 = 1 1</center> | <center>1 + 1 = 1 1</center> | ||
<center>1 1 + 1 1 = 1 5</center> | <center>1 1 + 1 1 = 1 5</center> | ||
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where the spaces have been added for clarity. | where the spaces have been added for clarity. | ||
− | + | The joke is that because Arabic numerals do not use the same rules of addition and subtraction as Roman numerals, simple concatenation makes the equations incorrect. For example, 11 is read as 10+1, not 1+1 as it should under the correct rules for interpreting Roman numerals. Randall derives additional humor from the premise that Cueball seems to know Roman numerals better than Arabic numerals (as demonstrated by the fact that he does not recognize that his equations are false when interpreted using the standard rules for Arabic numerals) so that he would do math in Roman numerals and have to remember to convert his equations to Arabic numerals at the end. Schoolchildren in the West have been taught to do math with Arabic numerals, not Roman numerals, for centuries. | |
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− | The joke is that because Arabic numerals do not use the same rules of addition and subtraction as Roman numerals, the equations | ||
In the title text, Randall applies the same idea of replacing Roman numerals with their values in Arabic numerals to strings of English words. | In the title text, Randall applies the same idea of replacing Roman numerals with their values in Arabic numerals to strings of English words. | ||
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− | The original string (with letters that would be interpreted as Roman numerals capitalized) is, "CheCk out thIs InnoVatIVe strIng enCoDIng I'Ve been DeVeLopIng! It's VIrtuaLLy perfeCt!" For the first word, "Check," C is replaced with the value of that Roman numeral in Arabic numerals, i.e., "100", in both instances | + | The original string (with letters that would be interpreted as Roman numerals capitalized) is, "CheCk out thIs InnoVatIVe strIng enCoDIng I'Ve been DeVeLopIng! It's VIrtuaLLy perfeCt!" For the first word, "Check," C is replaced with the value of that Roman numeral in Arabic numerals, i.e., "100", in both instances of the word, which results in "100he100k". Unlike in the comic, Randall combines Roman numbers using the proper rules of addition and subtraction. For example, he replaces "IV" with "4", not "15", e.g., "innovative" becomes "1nno5at4e", not "1nno5at15e". (However, "I've" becomes "15e", not "4e", presumably because the apostrophe was removed after, not before, replacing the Roman numerals with Arabic numerals. However, there is not an obvious reason why Randall removed the apostrophe.) However, there are problems with this. One example is that the double L in "virtually" is replaced with 100. This correctly remembers Roman numerals' rule of adding the value of a letter to the value of an equal-valued letter just to its right, but in Roman numerals, a single number should never have multiple Vs, multiple Ls, or multiple Ds, e.g., 100 should be represented by C, not LL. This would mean that a simplistic decoding script would erroneously decode "6rtua100y" to "virtuacy", not "virtually". Thus, this string encoding system is not actually perfect. (Until the modern codification in general use today, Roman numerals weren't standardised that much, so "LL" could have been a tolerated alternative to "C". For more on that, see {{w|Roman_numerals#Classical_Roman_numerals}}. However, having the decoding script use that would not solve the problem but instead would make the decoding script replace Cs with LLs instead, e.g., "delloding sllript".) |
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− | Until the modern codification in general use today, Roman numerals weren't standardised that much, so "LL" could have been a tolerated alternative to "C". For more on that, see {{w|Roman_numerals#Classical_Roman_numerals | ||
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==Transcript== | ==Transcript== | ||
− | :[Cueball | + | {{incomplete transcript|Do NOT delete this tag too soon.}} |
+ | :[Cueball writing on a wall or a whiteboard.] | ||
:1+1=11 | :1+1=11 | ||
:11+11=15 | :11+11=15 |