<P> Not only these prominent examples, but almost all real numbers are irrational and therefore have no repeating patterns and hence no corresponding decimal numeral . They can only be approximated by decimal numerals, denoting rounded or truncated real numbers . Any rounded or truncated number is necessarily a rational number, of which there are only countably many . All measurements are, by their nature, approximations, and always have a margin of error . Thus 123.456 is considered an approximation of any real number greater or equal to 1234555 / 10000 and strictly less than 1234565 / 10000 (rounding to 3 decimals), or of any real number greater or equal to 123456 / 1000 and strictly less than 123457 / 1000 (truncation after the 3 . decimal). Digits that suggest a greater accuracy than the measurement itself does, should be removed . The remaining digits are then called significant digits . For example, measurements with a ruler can seldom be made without a margin of error of at least 0.001 meters . If the sides of a rectangle are measured as 1.23 meters and 4.56 meters, then multiplication gives an area for the rectangle between 5.614591 square meters and 5.603011 square meters . Since not even the second digit after the decimal place is preserved, the following digits are not significant . Therefore the result is usually rounded to 5.61 . </P> <P> Just as the same fraction can be written in more than one way, the same real number may have more than one decimal representation . For example, 0.999..., 1.0, 1.00, 1.000,..., all represent the natural number 1 . A given real number has only the following decimal representations: an approximation to some finite number of decimal places, an approximation in which a pattern is established that continues for an unlimited number of decimal places, or an exact value with only finitely many decimal places . In this last case, the last non-zero digit may be replaced by the digit one smaller followed by an unlimited number of 9's, or the last non-zero digit may be followed by an unlimited number of zeros . Thus the exact real number 3.74 can also be written 3.7399999999...and 3.74000000000...Similarly, a decimal numeral with an unlimited number of 0's can be rewritten by dropping the 0's to the right of the decimal place, and a decimal numeral with an unlimited number of 9's can be rewritten by increasing the rightmost non-9 digit by one, changing all the 9's to the right of that digit to 0's . Finally, an unlimited sequence of 0's to the right of the decimal place can be dropped . For example, 6.849999999999...= 6.85 and 6.850000000000...= 6.85 . Finally, if all of the digits in a numeral are 0, the number is 0, and if all of the digits in a numeral are an unending string of 9's, you can drop the nines to the right of the decimal place, and add one to the string of 9s to the left of the decimal place . For example 99.999...= 100 . </P> <P> The real numbers also have an important but highly technical property called the least upper bound property . </P> <P> It can be shown that any ordered field, which is also complete, is isomorphic to the real numbers . The real numbers are not, however, an algebraically closed field, because they do not include a solution (often called a square root of minus one) to the algebraic equation x 2 + 1 = 0 (\ displaystyle x ^ (2) + 1 = 0). </P>

Explain number system and it's commonly used types