Decimal

What the point marks

The decimal point is not a place of its own. It is the boundary between whole units on its left and parts of one on its right.

$$3.47 = 3 + \tfrac{4}{10} + \tfrac{7}{100}$$

Decimals are fractions

Every decimal is a fraction whose denominator is a power of ten.

Decimal	Fraction
$0.4$	$\tfrac{4}{10}$
$0.07$	$\tfrac{7}{100}$
$0.375$	$\tfrac{375}{1000} = \tfrac{3}{8}$

That fixed denominator is the whole advantage. Because every decimal is built on tens, two of them can be compared place by place and added in columns — neither of which works on fractions until they have been given a common denominator.

Terminating and repeating

A fraction becomes a decimal that stops exactly when its simplified denominator is built only from $2$s and $5$s, because ten is $2 \times 5$.

$$\tfrac{3}{8} = 0.375 \qquad \tfrac{2}{3} = 0.\overline{6}$$

The bar marks digits that repeat forever. Any other prime factor — a $3$, a $7$ — never divides a power of ten, so the division never ends. See fractions to decimals.

Zeros

$$0.5 = 0.50 \qquad\text{but}\qquad 0.5 \ne 0.05$$

A zero on the end changes nothing. A zero between the point and the digits pushes every digit one place further right and divides the number by ten.

The name

From the Latin decem, ten — the same root as decade and December, which was the tenth month before the calendar shifted.