Roman arithmetic

In Rome, merchants used Roman numerals to perform basic arithmetic operations. In modern education, the Roman arithmetic used by the Romans is seldom taught. The prefered method is to convert the Roman numeral into an Arabic numeral and solve the equation using a modern positional notation system. While that's more practical, it is not really learning how to add, subtract, multiply and divide Roman numerals, it is only making the student practice converting from Roman to Arabic and back again. Except for historical purposes, none of this is particularly useful to the grade student unless it is used to demonstrate the existence of different numeral systems and their impact on Arithmetic and to do that, the student needs to learn how to perform arithmetic operations in the native numeral system.

Basic operations

All arithmetic operations can be broken down to combinations of addition and subtraction.

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Addition

Example

CXVI + XXIV = ?

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Solution:

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CXVI + XXIV = CXL

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Discussion

Step 1 decodes the positional data in the terms and replaces it with primitive counts. Now represented as a pure counting system, the concatenation of the terms in Step 2 gives the correct solution to the problem: CXVIXXIIII represents the same number as CXL - both terms convert to 140 in Arabic numerals. Steps 3 & 4 now reduce the result to the simplest expression possible and Step 5 reintroduces subtractive notation transforming the result back into a positional number.

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Subtraction

Example

CXVI − XXIV = ?

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Solution:

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CXVI − XXIV = XCII

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Discussion

Step 1 decodes the positional data in the terms and replaces it with primitive counts. In Step 2, like numerals are eliminated from both terms: a count of X and a count of I are each removed from each term, leaving a simplified problem of CV − XIII. Step 3 then expands the first term until it contains a common numeral (X) to the highest numeral in the second term. Step 2 is then repeated, followed by Step 3 until all of the numerals in the second term have been eliminated. Once all of the numerals of have been eliminated, the remaining numerals in the first term represent the solution as a primitive count. Step 5 reintroduces subtractive notation transforming the result back into a positional number.

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