Decimals

Decimals represent numbers using base-10 with a decimal point. In this system, digits to the left of the point form the integer part, while digits to the right form the fractional part, with each position corresponding to a negative power of ten (tenths, hundredths, thousandths, etc.). A rational number whose reduced denominator is a power of ten has a terminating decimal expansion, such as 0.75 for 75/100. If the denominator includes other primes, the decimal expansion is infinite and repeats periodically, such as 1/3 = 0.333...

Decimals can be converted to fractions by scaling by powers of ten, and fractions to decimals by division. Arithmetic with decimals follows base-10 rules: align decimal points for addition and subtraction, multiply as integers then place the decimal, and divide similarly or adjust decimal places. Decimal representations can be rounded or truncated to a required precision, introducing rounding error. Some decimals terminate (finite digits); others repeat with a repeating block (repeating decimals). A recurring decimal can be denoted with a bar over the repeating digits.

Decimals are a standard numeration form in science, engineering, and everyday use, especially for measurements and money. They complement fractions and percentages and are central to computer arithmetic, where floating-point representations approximate real numbers. The modern decimal system traces to early work in Europe, notably the work of Simon Stevin in the late 16th century, who popularized decimal fractions.