The Roman numeral system is a compact form of the tally method, and extends to somewhat large numbers, but it's fairly difficult to multiply and divide using Roman numerals. For that reason, Arabic numeral notation supplanted Roman numerals late in the Middle Ages.
Arabic notation is really a base 10 polynomial system. For example, the number six thousand, seven hundred, and eighty-three (6783) is six times the base (10) cubed, plus seven times the base squared, plus eight times the base, plus three times the base to the zeroeth power. By extending the polynomial system to negative exponents (using a decimal point), we can represent any rational number. Adding, subtracting, multiplying, and dividing are all straightforward using Arabic decimal notation. There are some tasks, however, for which this polynomial system is not well suited. Those tasks include factoring numbers and determining whether a number is prime.
Every positive whole number can be represented as a product of prime numbers.
Every whole number is either prime or composite. If it's prime, the product is trivial. If it's composite, the factors can themselves be represented as products of primes.
The first ten prime numbers are listed below:
The number 6783 can be represented as a product of the primes 3, 7, 17, and 19 (6783 = 3 x 7 x 17 x 19).
(0, 1, 0, 1, 0, 0, 1, 1)
The number 40, as a further example, would be represented like this:
(3, 0, 1)
because 40 is two cubed times five.
(0, 1, 0, 1, 0, 1) x (1, 2, 0, 0, 0, 1) ------------------ (1, 3, 0, 1, 0, 2)
Division is the reverse, simply subtract the vector components. If any of the components of the divisor are larger than a corresponding component of the dividend, then the dividend cannot be divided evenly by the divisor and there will be a remainder.
For example, this text file of the first million primes is about 20 megabytes. The format of the file is two columns, separated by commas. The first column is the series number of (nth) the prime, and the second column is the prime itself. For example, the hundredth prime is 541. The numbers in both columns are in quotation marks because they were saved as text strings from the generating program (Big Number Cruncher, a Windows 3.0 application). The file does not quite go up to a million entries.
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Last updated January 23, 2013, by Rick Wagner. Copyright © 2011-2013, all rights reserved.