When writing a number, each digit has a specific place or position linked to a value. This value is called the place value. As we write numbers in base |10,| each value associated with the positions is, in fact, a power of |10.|
Here is a table showing the main values corresponding to the position of the number. This is useful for understanding the place value of whole and integer numbers.
|
Name of the position |
Value |
Order of magnitude |
|---|---|---|
|
Hundred billions |
|100\ 000\ 000\ 000| |
Billions |
|
Ten billions |
|10\ 000\ 000\ 000| |
|
|
Billions |
|1\ 000\ 000\ 000| |
|
|
Hundred millions |
|100\ 000\ 000| |
Millions |
|
Ten millions |
|10\ 000\ 000| |
|
|
Millions |
|1\ 000\ 000| |
|
|
Hundred thousands |
|100\ 000| |
Thousands |
|
Ten thousands |
|10\ 000| |
|
|
Thousands |
|1\ 000| |
|
|
Hundreds |
|100| |
Ones |
|
Tens |
|10| |
|
|
Ones |
|1| |
Here is an example. The table below describes the different positions and place values in the following whole number. ||42\:567\:123||
|
Number |
|4| |
|2| |
|5| |
|6| |
|7| |
|1| |
|2| |
|3| |
|---|---|---|---|---|---|---|---|---|
|
Position |
Ten millions |
Millions |
Hundred thousands |
Ten thousands |
Thousands |
Hundreds |
Tens |
Ones |
|
Place value |
|\small 4 \times 10\:000\:000 \\ \small =\\ \small 40\:000\:000| |
|\small 2 \times 1\:000\:000 \\ \small =\\ \small 2\:000\:000| |
|\small 5 \times 100\:000\\ \small =\\ \small 500\:000| |
|\small 6 \times 10\:000 \\ \small =\\ \small 60\:000| |
|\small 7 \times 1\:000 \\ \small =\\ \small 2\:000| |
|\small 1 \times 100 \\ \small = \\ \small 100| |
|\small 2\times 10 \\ \small = \\ \small 20| |
|\small 3\times 1 \\ \small = \\ \small 3| |
For whole and natural numbers, the smallest place value is always the one.
For each shift by one position to the left, the value is |10| times larger than the previous one. This means the place value is multiplied by |10| with each shift to the left.
On the other hand, each change of a position to the right means the position’s value is |10| times smaller than the previous one. This means the place value is divided by |10| with each shift to the right.
Exponential notation can be used to simplify writing if there are a series of multiplications by the same quantity.
In the number |75 \: 489|, the place value of the number 7 is: ||7 \times 10 \: 000 = 70 \: 000||
Use exponential notation to get a simpler equivalent.
||\begin{align} 70 \: 000 &= 7 \times 10 \: 000 \\
&= 7 \times 10 \times 10 \times 10 \times 10 \\
&= 7 \times \underbrace{\color{blue}{10 \times 10 \times 10 \times 10}}_{\color{red}{4 \ \text{times}}} \\
&= 7 \times \color{blue}{10}^\color{red}{4} \end{align}||
Exponential notation can be used for any place value.
The place values of decimal numbers are similar to those of whole numbers. The only difference is the addition of positions to the right of the decimal point. The decimal point is what separates the whole part from the decimal part.
The following table represents the main values associated with the positions of different digits in decimals.
|
Position Name |
Value |
Order of Magnitude |
|---|---|---|
|
Hundred Billions |
|100\ 000\ 000\ 000| |
Billions |
|
Ten Billions |
|10\ 000\ 000\ 000| |
|
|
Billions |
|1\ 000\ 000\ 000| |
|
|
Hundred Millions |
|100\ 000\ 000| |
Millions |
|
Ten millions |
|10\ 000\ 000| |
|
|
Millions |
|1\ 000\ 000| |
|
|
Hundred Thousands |
|100\ 000| |
Thousands |
|
Ten Thousands |
|10\ 000| |
|
|
Thousands |
|1\ 000| |
|
|
Hundreds |
|100| |
Ones |
|
Tens |
|10| |
|
|
Ones |
|1| |
|
|
Decimal Point |
. |
Separator |
|
Tenths |
|0.1| or |\dfrac{1}{10}| |
For the decimal part of a number, each position corresponds to an order of magnitude. |
|
Hundredths |
|0.01| or |\dfrac{1}{100}| |
|
|
Thousandths |
|0.001| or |\dfrac{1}{1\,000}| |
|
|
Ten thousandths |
|0.000\,1| or |\dfrac{1}{10\,000}| |
|
|
Hundred thousandths |
|0.000\,01| or |\dfrac{1}{100\,000}| |
|
|
Millionths |
|0.000\,001| or |\dfrac{1}{1\,000\,000}| |
Here is an example. The table below describes the different positions and place values in the following decimal number. ||54\:782 913||
|
Number |
|5| |
|4| |
|7| |
|8| |
|2| |
|\Large .| |
|9| |
|1| |
|3| |
|---|---|---|---|---|---|---|---|---|---|
|
Position |
Ten thousands |
Thousands |
Hundreds |
Tens |
Ones |
|
Tenths |
Hundredths |
Thousandths |
|
Place value |
|\small 5 \times 10\:000 \\ \small = \\ \small 50\:000| |
|\small 4\times 1\:000 \\ \small = \\ \small 4 \: 000| |
|\small 7\times 100 \\ \small = \\ \small 700| |
|\small 8\times 10 \\ \small = \\ \small 80| |
|\small 2\times 1 \\ \small = \\ \small 2| |
|
|\small 9\times 0.1 \\ \small = \\ \small 0.9| |
|\small 1\times 0.01 \\ \small = \\ \small 0.01| |
|\small 3\times 0.001 \\ \small = \\ \small 0.003| |
As mentioned previously, each decimal place value can be represented by a fraction.
For example, the value of the hundredth position corresponds to one over one hundred. ||0.01=\dfrac{1}{100}||
Just like with natural numbers and integers, exponential notation can be used to simplify the writing of the place value of the decimal portion.
In the number |75.489,| the place value of the digit |9| is: ||9 \times 0.001 = 0.009||
Here is the equivalent with exponential notation. ||\begin{align} 0.009 &= 9 \times 0.001 \\
&=9 \times \frac{1}{1\:000} \\
&= 9 \times \frac{1}{10} \times \frac{1}{10} \times \frac{1}{10}\\
&= 9 \times \underbrace{\color{blue}{\frac{1}{10} \times \frac{1}{10} \times \frac{1}{10}}}_{\color{red}{3 \ \text{times}}} \\
&= 9 \times \frac{\color{blue}{1}}{\color{blue}{10}^\color{red}{3}} \\
&= 9 \times \color{blue}{10}^\color{red}{\text{-}3}\end{align}||
Exponential notation can be used for any place value.