When measuring objects with high precision, standard units like centimeters may not be sufficient. We often encounter lengths that fall between two consecutive centimeter marks on a ruler, necessitating the use of smaller, subdivided units.
When a single unit is divided into $10$ equal parts, each part is called one-tenth of a unit, represented mathematically as $\frac{1}{10}$.
To measure even smaller lengths, we can divide each one-tenth part into $10$ equal subdivisions. Since there are $10$ tenths in a unit, dividing each tenth into $10$ parts yields $100$ equal parts in a single unit.
The decimal system is a natural extension of our base-10 place value system to represent fractional parts. Each place value is $10$ times smaller than the one immediately to its left.
To separate the whole number (integer) part from the fractional part, we use a separator called the decimal point (represented by a period '.').
The place values extend to the right of the ones place as follows:
Decimal notation simplifies conversions between different units of length, weight, and currency.
To locate a decimal number like $1.4$ on a number line, we identify that it lies between $1$ and $2$. We divide the unit interval between $1$ and $2$ into $10$ equal parts and count $4$ parts from $1$ to find $1.4$.
Appending zeros to the rightmost end of a decimal number does not change its value. For example: $$0.2 = 0.20 = 0.200$$ This is because they all represent the same quantity ($2$ tenths). However, $0.2$, $0.02$, and $0.002$ are entirely different values because the digit $2$ occupies different place values.
To compare two decimal numbers:
Decimals are added and subtracted using the same standard vertical algorithms as whole numbers, with one critical rule: align the decimal points vertically.
Sometimes, decimal-like notations are used in non-decimal contexts, which can lead to confusion:
Decimal fractions were used by ancient Indian astronomers and mathematicians, including the 8th-century scholar Śhrīdharāchārya. The system was detailed in the 10th century by Arab mathematician Abūl Ḥassan al-Uqlīdisī. In the 16th century, mathematicians like John Napier popularized the use of the point/period ('.') as a decimal separator.