Kümnendvorming

Kümnendvorming is the process of expressing numbers in decimal form, using base-10 positional notation. In this system, numbers are written with a decimal point separating the integer part from the fractional part. Each digit to the right of the decimal point represents tenths, hundredths, thousandths, and so on.

Decimal forms can be terminating or repeating. A fraction a/b converts to a decimal by division. If the denominator b, after reducing the fraction, has only the prime factors 2 and 5, the decimal terminates. If other primes occur, the decimal repeats indefinitely.

To obtain a terminating decimal, one can multiply numerator and denominator by a suitable power of 10 so that the denominator becomes 10^k, resulting in a decimal with k digits after the decimal point. For repeating decimals, the decimal expansion continues with a repeating block of digits (the repetend) after a certain point.

Examples illustrate the distinction. 3/4 equals 0.75 (terminating). 1/3 equals 0.333… (repeating). 1/8 equals 0.125 (terminating). 7/20 equals 0.35 (terminating). In many contexts, decimals are rounded to a given number of digits, affecting precision and error.

Historically, decimal fractions and decimal notation were promoted in Europe in the early modern period, notably by Simon Stevin, which facilitated arithmetic, measurement, and science. Today, decimal formation is a standard tool in education, science, finance, and engineering.