Decimal Numbers
Rational expressions whose denominators consist of positive integer powers of 10, or those that can be converted into this format through expansion or simplification, are called decimal numbers.
$$ \frac{7}{10} = 0.7 $$
$$\frac{3}{10^2} = 0.03 $$
Place Values and Digit Values of Decimals:
The digit positions in a decimal expression separated by a comma (or decimal point) are mathematically named based on whether they reside in the whole-number part on the left or the fractional part on the right:
Let’s write the place values of the digits in the number $127.908$.
$$
\begin{array}{|l|c|}
\hline
\text{Place Value Name} & \text{Digit} \\ \hline
\text{Hundreds Place} & 1 \\ \hline
\text{Tens Place} & 2 \\ \hline
\text{Ones Place} & 7 \\ \hline
\text{Tenths Place} & 9 \\ \hline
\text{Hundredths Place} & 0 \\ \hline
\text{Thousandths Place} & 8 \\ \hline
\end{array}
$$
Let’s write the expanded form of the same number.
$$ 127.908 = 1 \cdot 100 + 2 \cdot 10 + 7 \cdot 1 + 9 \cdot 0.1 + 0 \cdot 0.01 + 8 \cdot 0.001 $$
$$ = 100 + 20 + 7 + 0.9 + 0.00 + 0.008 $$
$$ = 1 \cdot 100 + 2 \cdot 10 + 7 \cdot 1 + 9 \cdot \frac{1}{10} + 0 \cdot \frac{1}{100} + 8 \cdot \frac{1}{1000} $$
All three presentation models provided above in different formats are variations of the expanded form of the number $127.908$. In summary, expressing a decimal number as the sum of the individual place values of each digit that comprises it is called the expansion of a decimal fraction.