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We began our study of arithmetic ( [link] ) by noting that our number system is called a positional number system with base ten. We also noted that each position has a particular value. We observed that each position has ten times the value of the position to its right.
This means that each position has $\frac{1}{10}$ the value of the position to its left.
Thus, a digit written to the right of the units position must have a value of $\frac{1}{\text{10}}$ of 1. Recalling that the word "of" translates to multiplication $\left(\cdot \right)$ , we can see that the value of the first position to the right of the units digit is $\frac{1}{\text{10}}$ of 1, or
$\frac{1}{\text{10}}\cdot 1=\frac{1}{\text{10}}$
The value of the second position to the right of the units digit is $\frac{1}{\text{10}}$ of $\frac{1}{\text{10}}$ , or
$\frac{1}{\text{10}}\cdot \frac{1}{\text{10}}=\frac{1}{{\text{10}}^{2}}=\frac{1}{\text{100}}$
The value of the third position to the right of the units digit is $\frac{1}{\text{10}}$ of $\frac{1}{\text{100}}$ , or
$\frac{1}{\text{10}}\cdot \frac{1}{\text{100}}=\frac{1}{{\text{10}}^{3}}=\frac{1}{\text{1000}}$
This pattern continues.
We can now see that if we were to write digits in positions to the right of the units positions, those positions have values that are fractions. Not only do the positions have fractional values, but the fractional values are all powers of 10 $\left(\text{10},{\text{10}}^{2},{\text{10}}^{3},\dots \right)$ .
Notice that decimal numbers have the suffix "th."
The following numbers are examples of decimals.
The 6 is in the tenths position.
$\text{42}\text{.}6=\text{42}\frac{6}{\text{10}}$
The 8 is in the tenths position.
The 0 is in the hundredths position.
The 1 is in the thousandths position.
The 4 is in the ten thousandths position.
$9\text{.}\text{8014}=9\frac{\text{8014}}{\text{10},\text{000}}$
The 9 is in the tenths position.
The 3 is in the hundredths position.
$0\text{.}\text{93}=\frac{\text{93}}{\text{100}}$
The 7 is in the tenths position.
$0\text{.}7=\frac{7}{\text{10}}$
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