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Decomposing numbers is examining how numbers are built using the place values of the digits they contain. When the composition of the numbers is understood, they can, for example, be placed in order. This concept sheet will focus on how to decompose a number.
Each type of decomposition will include an example with a natural number and a decimal number.
The decomposition presented here should not be confused with factoring numbers.
Factoring numbers, often called multiplicative decomposition, is examining the various factors and prime factors of numbers.
This method is used to become familiar with the decomposition of numbers. It consists of using words to identify the position of each digit in the number.
Natural number
Decompose the number |91\:245| using place value names.
To clearly identify the position of the digits that make up the number, a table like the following can be used.
|
Digit |
|9| |
|1| |
|2| |
|4| |
|5| |
|---|---|---|---|---|---|
|
Position |
Ten thousands |
Thousands |
Hundreds |
Tens |
Ones |
Thus, the decomposition of the number |91\:245| contains |9| ten thousands, |1| thousand, |2| hundreds, |4| tens, and |5| ones.
Decimal number
Decompose the number |3.208| using place value names.
To keep track of the position of the digits that make up this number, it is possible to use a table like the following.
|
Number |
|3| |
|\large ,| |
|2| |
|0| |
|8| |
|---|---|---|---|---|---|
|
Position |
Ones |
|
Tenths |
Hundredths |
Thousandths |
Thus, the decomposition of the number |3.208| contains |3| ones, |2| tenths, and |8| thousandths.
As the previous example illustrates, certain positions contain the number |0.| Since the value is zero, the decomposition will contain fewer notations.
Note: since we have the number |0| in the hundreds position, it is not necessary to include it in the decomposition.
The number is decomposed by adding the place values of all the digits that make it up.
To determine the place value of the digits, isolate each digit and replace all the digits between it and the decimal point with |0|.
For example, to find the place value of the digit |\color{blue}{5}| in the number |1\:541.221| , proceed as follows.
||1\:\color{blue}{541.}221\ \mapsto\ \not 1 \ \color{blue}{541.} \not 2 \not 2 \not 1 \ \mapsto \ \color{blue}{500}||
Therefore, the place value of the digit |\color{blue}{5}| is |500| .
All numbers can be quickly decomposed using the above trick.
Natural number
Decompose the number |91\:245| using additive decomposition.
||\begin{align}\color{blue}{91\:245}\ &\mapsto\ \color{blue}{90\:000}\\
9\color{red}{1\:245}\ &\mapsto\ \color{red}{1\:000}\\ 91\:\color{green}{245}\ &\mapsto\ \color{green}{200}\\ 91\:2\color{fuchsia}{45}\ &\mapsto\ \color{fuchsia}{40}\\ 91\:24\color{orange}{5}\ &\mapsto\ \color{orange}{5}\end{align}||
Therefore, the additive decomposition of the number is: ||91\:245=\color{blue}{90\:000}+\color{red}{1\:000}+\color{green}{200}+\color{fuchsia}{40}+\color{orange}{5}||
Decimal number
Decompose the number |3.208| using additive decomposition.
Still using the trick above:
||\begin{align}\color{blue}{3.}208\ &\mapsto\ \color{blue}{3}\\
3\color{red}{.2}08\ &\mapsto\ 0.\color{red}{2}\\ 3\color{green}{.20}8\ &\mapsto\ 0.\color{green}{00}\\ 3\color{fuchsia}{.208}\ &\mapsto\ 0.\color{fuchsia}{008}\end{align}|| Therefore, the additive decomposition of the number is: ||3.208=\color{blue}{3}+\color{red}{0.2}+\color{fuchsia}{0.008}||
Note: When the digit |0| is in a number, the place value of the digit is equal to |0|. It is not useful to include it in the additive decomposition.
In the previous example, it is the |\color{green}{\text{green}}| part.
For the additive decomposition of decimal numbers, it is possible to represent the place values of the decimals by decimal fractions.
In the previous example, the decomposition would be written as follows. ||3.208=\color{blue}{3}+\color{red}{\frac{2}{10}}+\color{fuchsia}{\frac{8}{1000}}||
The following decomposition method is very similar to the additive method. The difference is that each place value is represented by multiplying the digit and the value associated with its position.
To perform this type of decomposition, it is important to know the values associated with the positions.
Natural number
Decompose the number |91 \: 245| in expanded form.
By using the values associated with the positions of the digits, we obtain the following expanded form. ||91\:245=(9\times 10\:000)+(1\times 1\:000)+(2\times 100)+(4\times 10)+(5\times 1)||
Decimal number
Decompose the number |3.208| in expanded form.
By using the values associated with the positions of the digits, we obtain the following expanded form. ||\begin{align} 3.208&=(3\times 1)+(2\times 0.1)+(8\times 0.001)\\
&=(3\times 1)+\left(2\times \frac{1}{10}\right)+\left(8\times \frac{1}{1000}\right)\end{align}||
Exponential notation can be used to condense the expanded form.
Decompose the number |91 \: 245| in expanded form.
||\begin{align} 91\:245&=\color{blue}{9\times 10\:000}+\color{red}{1\times 1\:000}+\color{green}{2\times 100}+\color{fuchsia}{4\times 10}+\color{orange}{5\times 1} \\
&=\color{blue}{9\times 10^4}+\color{red}{1\times 10^3}+\color{green}{2\times 10^2}+\color{fuchsia}{4\times 10^1}+\color{orange}{5\times 10^0}\\
&=\color{blue}{9\times 10^4}+\color{red}{1\times 10^3}+\color{green}{2\times 10^2}+\color{fuchsia}{4\times 10}+\color{orange}{5\times 1}\end{align}||