When comparing natural numbers. it is possible to sort them in relation to one another. They can then be placed in ascending or descending order.
Sorting numbers in ascending order means placing them from smallest to largest.
|0\ <\ 1\ <\ 2\ <\ 3\ <\ 4\ <\ 5\ <\ 6\ <\ ...|
Sorting numbers in descending order means placing them from largest to smallest.
|...\ >\ 6\ >\ 5\ >\ 4\ >\ 3\ >\ 2\ >\ 1\ >\ 0||
The order of natural numbers can be represented in several ways. Following are two methods for doing so.
The order of small natural numbers can be represented using drawings. The more drawings (of the same value) necessary to represent a number, the greater the number.
The following numbers are represented using stars.
We can see that |\small 1| is the smallest number, while |\small 12| is the biggest.
The ascending order of the numbers is:
||1\ <\ 4\ <\ 7\ <\ 12||
The descending order is:
||12\ >\ 7\ >\ 4\ >\ 1||
We can also use a number line to represent the order of numbers.
The farther away a natural number is from |\small 0| on the right, the greater the number will be.
The following numbers are represented on the number line.
We can see that |\small 6| is the whole natural number with the smallest value, because it is the one that is closest to |\small 0.| Using the same logic, the whole natural number |\small 34| has the greatest value, because it is furthest from |\small 0.|
The ascending order of the numbers is:
||6\ <\ 13\ <\ 20\ <\ 34||
The descending order of the numbers is: ||34\ >\ 20\ >\ 13\ >\ 6||
Several methods allow you to order natural numbers. We will present two of them.
When natural numbers must be ordered, they can be separated into groups according to the number of digits they have. This makes it easier to sort them. The following are the steps in the first method.
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Determine whether to place the numbers in ascending or descending order.
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Group the numbers to be sorted according to the number of digits they have.
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Sort the numbers in each group according to the desired order.
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Put the groups together to get the numbers in the desired order.
To sort natural numbers consisting of the same number of digits, you must compare the digits with the same place values together, starting from left to right. As soon as there is a difference, the larger number will belong to the larger number.
For example, the numbers |1245| and |1263.|
||\large \bbox[orange]{1}\: \bbox[aqua]{2}\: \bbox[yellow]{4}\:\bbox[pink]{5}\qquad \qquad \bbox[orange]{1}\: \bbox[aqua]{2}\: \bbox[yellow]{6}\:\bbox[pink]{3}||
Start by comparing the digits in the leftmost position – the one with units of thousand. We have the number |\bbox[orange]{1}| for the two numbers, therefore, we must look at the digits in the hundreds position.
Note that the two numbers are the same, |\bbox[aqua]{2}| in the hundreds position as well. We continue towards the right by comparing the digits in the tens position.
In this position, we notice a difference, since |\bbox[yellow]{6}| is larger than |\bbox[yellow]{4},| and the number |12\bbox[yellow]{6}5| is larger than |12\bbox[yellow]{4}5|. The result is: ||\large 12\bbox[yellow]{4}5\ <\ 12\bbox[yellow]{6}3||
Place the following numbers in ascending order:
||1\qquad 26\qquad 859\qquad 56 \qquad 8\qquad 24\qquad 347||
1. Determine whether to place the numbers in ascending or descending order
As mentioned in the question, the numbers should be placed in ascending order, from smallest to largest.
2. Group the numbers to be ordered according to the number of digits they have.
||\enclose{circle}[mathcolor="green"]{\color{black}{1}}\qquad \enclose{circle}[mathcolor="blue"]{\color{black}{26}}\qquad \enclose{circle}[mathcolor="purple"]{\color{black}{859}}\qquad \enclose{circle}[mathcolor="blue"]{\color{black}{56}} \qquad \enclose{circle}[mathcolor="green"]{\color{black}{8}}\qquad \enclose{circle}[mathcolor="blue"]{\color{black}{24}}\qquad \enclose{circle}[mathcolor="purple"]{\color{black}{347}}||||\underbrace{\color{green}{1}\ , \ \color{green}{8}}_\text{one digit}\qquad\qquad \underbrace{\color{blue}{26}\ ,\ \color{blue}{56}\ ,\ \color{blue}{24}}_\text{two digits}\qquad\qquad \underbrace{\color{purple}{859}\ ,\ \color{purple}{347}}_\text{three digits}||
3. Sort the numbers in order for each group.
Using the trick above, we can sort the numbers in each of the three groups we have created, in ascending order.
This is the result:
||\color{green}{1}\ <\ \color{green}{8}\qquad \qquad \color{blue}{24}\ <\ \color{blue}{26}\ <\ \color{blue}{56}\qquad \qquad \color{purple}{347}\ <\ \color{purple}{859}||
4. Put the groups together
Since a number consisting of two digits is greater than a number consisting of a single digit, we get the following:
||\color{green}{1}\ <\ \color{green}{8}\ <\ \color{blue}{24}\ <\ \color{blue}{26}\ <\ \color{blue}{56} \ <\ \color{purple}{347}\ <\ \color{purple}{859}||
If the previous example’s instructions had been to sort the numbers in descending order, it would have been sufficient to reverse the sequence in step 4.
||\color{purple}{859}\ >\ \color{purple}{347}\ >\ \color{blue}{56}\ >\ \color{blue}{26}\ >\ \color{blue}{24}\ >\ \color{green}{8}\ >\ \color{green}{1}||
Natural numbers can be placed using a number line. Here are the steps for this second method.
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Determine whether to place the numbers in ascending or descending order.
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If it is not already provided, draw a number line.
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Taking into account the graduation mark, position the numbers to be sorted on the number line.
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Sort the numbers in the desired order.he further a number is from zero, the larger it is.
The graduation mark of a number line is the difference between two consecutive marks.
Place the following numbers in descending order:
||340\qquad 155\qquad 210\qquad 100\qquad 275\qquad 410||
1. Determine whether to place the numbers in ascending or descending order.
As stated in the question, the numbers should be placed in descending order, from largest to smallest.
2. Draw a number line.
We decide to draw a number line with a graduation mark of |50|.
3. Taking into account the graduation mark, position the numbers to be sorted on the numerical line.
4. Sort the numbers in the desired order.
Since the largest numbers are the furthest from zero, the result is:
||410\ >\ 340\ >\ 275\ >\ 210\ >\ 155\ >\ 100||
If the previous example’s instructions had been to sort the numbers in ascending order, it would have been sufficient to reverse the sequence found in step 4.
||100\ <\ 155\ <\ 210\ <\ 275\ <\ 340\ <\ 410||