Ordering Integers Ascending and Descending

Hello again. In the previous lesson, you compared two integers by using the number-line rule: values farther right are greater, and values farther left are less. Ordering is the same idea applied to a whole collection of integers.

By the end of this lesson, you will be able to arrange integers in ascending order (least to greatest) and descending order (greatest to least), whether or not you draw a number line. This is the final skill in this module; it brings together positive numbers, negative numbers, zero, absolute value, and comparison.

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Two directions for ordering

A number line gives the meaning of both types of order:

• Ascending order means from the least value to the greatest value.
• Descending order means from the greatest value to the least value.

For example, consider the integers

$$-4,2,0,-1,5$$

In ascending order, begin with the value farthest left:

$$-4,-1,0,2,5$$

You can also show ascending order with inequality signs:

$$-4<-1<0<2<5$$

For descending order, use the same values in the opposite direction:

$$5,2,0,-1,-4$$

Or:

$$5>2>0>-1>-4$$

The words in the question tell you which direction to read:

If the question says...	Start with...	Move on the number line...
ascending, least to greatest, smallest to largest	the least integer	left to right
descending, greatest to least, largest to smallest	the greatest integer	right to left

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A dependable method for a mixed set of integers

You do not always need to draw a number line, but you should be able to picture one. For a mixed group of positive and negative integers, organize the numbers into three groups:

1. Negative integers
2. Zero, if it appears
3. Positive integers

All negative integers are less than zero, and zero is less than all positive integers. So, in ascending order, the negatives must come first; in descending order, the positives must come first.

The part requiring the most care is sorting the negatives.

Sorting negative integers

For negative integers:

• The number farther from zero is smaller.
• The number closer to zero is greater.

For instance:

$$-12<-7<-3$$

Although $12$ is greater than $7$, negative twelve lies farther left on the number line, so it is the smaller integer.

Here is a useful pattern:

Negative integers	Ascending order	Descending order
$-9,-2,-5$	$-9,-5,-2$	$-2,-5,-9$

For ascending order among negatives, begin with the negative having the greatest absolute value. For descending order among negatives, begin with the negative closest to zero.

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Worked example: ascending order

Arrange the following integers from least to greatest:

$$6,-8,0,-3,4,-1$$

First, separate the values mentally:

• Negatives: $-8,-3,-1$
• Zero: $0$
• Positives: $4,6$

Next, sort the negatives. The farthest left is $-8$, followed by $-3$, then $-1$.

Finally, place zero after the negatives and sort the positives normally:

$$-8,-3,-1,0,4,6$$

A quick check is to read the list from left to right as a chain:

$$-8<-3<-1<0<4<6$$

Every comparison is true, so the ordering is correct.

The diagram shows an important habit: when a number line is available, plot each number and simply read the points in the direction requested. For ascending order, read left to right.

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Worked example: descending order

Now arrange this set from greatest to least:

$$-6,3,0,-2,8,5$$

Because this is descending order, start with the greatest values. The positive integers come first, arranged from greatest to least:

$$8,5,3$$

Next comes zero:

$$8,5,3,0$$

The negative integers come last. Among negatives, the one closest to zero is greatest, so $-2$ comes before $-6$:

$$8,5,3,0,-2,-6$$

Check the direction with inequality signs:

$$8>5>3>0>-2>-6$$

Notice that descending order is not “put all the numbers backward” without thinking. It means each new number must be less than or equal to the previous one.

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Ordering integers in context

Ordering tells a story when integers represent real quantities.

Suppose these are morning temperatures in degrees:

$$-10,3,-4,0,7$$

From coldest to warmest, the temperatures are in ascending order:

$$-10,-4,0,3,7$$

From warmest to coldest, they are in descending order:

$$7,3,0,-4,-10$$

The negative temperatures are not ordered by their digits alone. A temperature of $-10$ degrees is colder, and therefore smaller, than $-4$ degrees.

The same reasoning works with elevations. For example, elevations of $-50$ meters, $-8$ meters, $0$ meters, and $15$ meters are ordered from lowest to highest as:

$$-50,-8,0,15$$

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Avoid these common mistakes

Ordering negatives as though they were positive

This is incorrect:

$$-2,-5,-9$$

as an ascending list, because $-2$ is actually the greatest of the three. The correct ascending order is:

$$-9,-5,-2$$

Forgetting that zero has a position

Zero is neither positive nor negative, but it belongs between them:

$$-4<0<6$$

Do not leave it out or place it before a negative integer in ascending order.

Using absolute value as the final order

Absolute value measures distance from zero, not whether a number is greater. For example:

$$|-8|=8$$

and

$$|5|=5$$

But that does not mean $-8$ is greater than $5$. In fact:

$$-8<5$$

Absolute value is only a helpful shortcut when comparing two negative integers.

Dropping a repeated integer

If a list has the same integer more than once, include every occurrence. For example, ordering

$$-2,4,-2,0$$

in ascending order gives:

$$-2,-2,0,4$$

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A final checking routine

After ordering a set, use these three checks:

1. Count: Did every integer from the original list appear once in your answer?
2. Direction: Does each number get greater in ascending order, or smaller in descending order?
3. Negatives: Did you remember that among negatives, values closer to zero are greater?

For instance, if an ascending list ends

$$-3,-7$$

the order is wrong, because $-7$ is less than $-3$. They should appear as:

$$-7,-3$$

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Key takeaways

• Ascending order means least to greatest: read a number line from left to right.
• Descending order means greatest to least: read a number line from right to left.
• In ascending order, negative integers come first, then zero, then positive integers.
• Among negative integers, the one farther left is smaller:

$$-11<-4<-1$$

• Check your result by reading adjacent numbers with $<$ for ascending order or $>$ for descending order.

Next, you will begin calculating with integers by adding two integers with the same sign.