Comparing and Ordering Integers | Grade 8 Math | LevelKite

You already know that $5$ is greater than $2$ . In this lesson you will make the same kind of comparison when some of the numbers are below zero.

The integers are the counting numbers $1, 2, 3, \dots$ , their negatives $-1, -2, -3, \dots$ , and $0$ . The counting numbers are positive, the integers written with a minus sign are negative, and $0$ is neither.

To compare two integers is to decide which one is less and which one is greater, or whether they are equal. To order a list of different integers is to arrange it so that each number is less than the next (least to greatest) or greater than the next (greatest to least).

When two integers are different, you write the comparison with $<$ , which means is less than, or $>$ , which means is greater than. The open side of the symbol faces the greater number, so $2 < 5$ and $5 > 2$ state the same fact.

What does less mean below zero? Think of a thermometer that measures degrees Celsius, where $0$ is the temperature at which water freezes. A morning at $-10$ degrees is colder than a morning at $-2$ degrees. The colder temperature is the lesser number, so $-10$ is less than $-2$ , even though $10$ is greater than $2$ .

A number line turns this idea into positions. It is a straight line with equally spaced marks called ticks. Zero is the reference point. The positive integers sit to the right of $0$ , one at each tick, and the negative integers sit to the left of $0$ in steps of the same size. Each integer sits exactly on its own tick, so its position is exact, not an estimate.

Values on the line increase as you move right: each step to the right lands on a greater integer, and each step to the left lands on a lesser one. The figure below shows three integers on such a line.

From left to right the markers sit at

-6

,

-1

and

3

, so

-6 < -1 < 3

.

The line above shows the part from $-8$ to $8$ in steps of $1$ . The full line continues without end in both directions. Each marker is exactly on a tick, so you can read its value instead of judging by eye: the markers sit at $-6$ , $-1$ and $3$ .

Reading from left to right, you meet $-6$ , then $-1$ , then $3$ . So $-6$ is the least of the three and $3$ is the greatest. As a chain, $-6 < -1 < 3$ : each number is less than the one after it.

An integer's distance from zero is the number of steps of $1$ from its tick to $0$ . Here $-6$ is $6$ steps from $0$ and $3$ is only $3$ steps from $0$ , yet $-6$ is the lesser number. Distance alone does not decide the order. You also need to know which side of $0$ the integer is on, and its sign, positive or negative, tells you that: negative integers are on the left and positive integers are on the right.

Key concept

Left is less, right is greater

On a number line whose values increase from left to right, position decides every comparison of two integers:

the integer farther left is less;
the integer farther right is greater;
two integers at the same tick are equal.

For any integer other than $0$ , the sign gives the side of $0$ and the distance from zero gives how far along that side it sits. Together they fix the exact position, so you can compare integers without drawing the line.

Comparing two different integers by sign and distance from zero

The two integers	Where they sit	Which is less	Example
One negative, one positive	The negative one is left of $0$ ; the positive one is right of $0$ .	The negative integer	$-9 < 2$
One negative, the other $0$	The negative one is left of $0$ .	The negative integer	$-4 < 0$
One positive, the other $0$	The positive one is right of $0$ .	$0$	$0 < 6$
Both positive	The one farther from $0$ is farther right.	The one closer to $0$	$4 < 7$
Both negative	The one farther from $0$ is farther left.	The one farther from $0$	$-7 < -4$

Worked example

Compare two negative integers

Which is greater, $-8$ or $-3$ ? Write the comparison with $<$ and with $>$ .

Check the signs. Both integers are negative, so both sit to the left of 0. The last row of the table applies.
Find each distance from zero. The integer −8 is 8 steps to the left of 0, and −3 is 3 steps to the left of 0.
Compare the positions. Because 8 steps is farther than 3 steps, −8 sits farther left than −3. On a number line whose values increase to the right, farther left means less.
$-8 < -3$
State the same fact starting with the greater number. Swap the two numbers and turn the symbol around so that its open side still faces −3. The meaning does not change.
$-3 > -8$

Answer

The greater integer is $-3$ . In symbols, $-8 < -3$ , which says the same as $-3 > -8$ .

The line runs from

-9

to

9

. The pairs

3

,

8

and

-3

,

-8

are the same distances from

0

, but

3 < 8

while

-8 < -3

.

Worked example

Order a list from least to greatest

Order these integers from least to greatest: $3$ , $-9$ , $0$ , $-2$ , $5$ , $-6$ .

Group the integers by their side of 0. The negatives are −9, −2 and −6, and the positives are 3 and 5. Every negative integer is left of 0 and every positive integer is right of 0, so the negatives come first, then 0, then the positives.
Order the negatives. Their distances from zero are 9, 2 and 6 steps. For negative integers, farther from zero means farther left, so the integer 9 steps away comes first, then the one 6 steps away, then the one 2 steps away.
$-9 < -6 < -2$
Order the positives. For positive integers, farther from zero means farther right, and 3 is closer to 0 than 5 is, so 3 comes before 5.
$3 < 5$
Join the groups in the order negatives, zero, positives. Each number in the chain is less than the one after it.
$-9 < -6 < -2 < 0 < 3 < 5$
Check with the number line below, which runs from −10 to 6 in steps of 1. Read from left to right, its markers sit at −9, −6, −2, 0, 3 and 5, which matches the chain.

Answer

From least to greatest, the integers are $-9$ , $-6$ , $-2$ , $0$ , $3$ , $5$ .

Read from left to right, the markers give the order from least to greatest:

-9

,

-6

,

-2

,

0

,

3

,

5

.

Worked example

Order temperatures from warmest to coldest

Five towns report their lowest temperature for the same night, in degrees Celsius: Town A $-12$ , Town B $4$ , Town C $-7$ , Town D $0$ and Town E $-15$ . List the temperatures from warmest to coldest. Which town was coldest?

Turn the question into a comparison. On the Celsius scale a warmer temperature is a greater number, so warmest to coldest means greatest to least.
Place the positive value and zero. The only positive value is 4. A positive integer is greater than 0, and 0 is greater than every negative integer, so the list starts with 4, then 0.
$4 > 0$
Order the negative values. Their distances from zero are 12, 7 and 15 steps. For negative integers, closer to zero means farther right, so it means greater. Take them from closest to farthest.
$-7 > -12 > -15$
Join the parts into one chain in which each number is greater than the one after it.
$4 > 0 > -7 > -12 > -15$
Match the least number to its town. The chain ends with −15, the temperature in Town E.

Answer

From warmest to coldest, the temperatures in degrees Celsius are $4$ , $0$ , $-7$ , $-12$ , $-15$ , so Town E was coldest.

To compare or order integers, think about position. On a number line whose values increase from left to right, the integer farther left is less and the integer farther right is greater. That is why every negative integer is less than $0$ and $0$ is less than every positive integer. It is also why, for two negative integers, the one farther from zero is the lesser one: $-8 < -3$ even though $8 > 3$ .

Next, you will add and subtract integers and use them to solve word problems.