Integers and Number Relationships
2. Comparing and Ordering Integers
Learning outcomes
- I can compare integers using a number line.
- I can order integers from least to greatest.
- I can order integers from greatest to least.
- I can explain why negative numbers are less than positive numbers.
- I can solve problems involving integer comparisons.
What Does It Mean to Compare Integers?
To compare integers means to determine which integer has the greater or smaller value.
For example:
7 > 3
because 7 is greater than 3.
But integers can also be negative:
−2 > −6
This can seem less obvious at first.
A number line provides one of the best ways to understand integer comparisons.
Review: What Are Integers?
Integers include:
- positive whole numbers
- negative whole numbers
- zero
For example:
..., −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, ...
Numbers such as:
2.5, −3.7, 1/2
are not integers.
The Integer Number Line
A number line places integers according to their value.
For example:
−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6
The most important rule is:
Numbers increase as you move to the right.
Numbers decrease as you move to the left.
Therefore:
The integer farther to the right is greater.
Comparison Symbols
Three symbols are commonly used.
> means greater than
Example:
5 > 2
< means less than
Example:
−4 < 3
= means equal to
Example:
−7 = −7
The open side of the comparison symbol faces the greater value.
Comparing Positive Integers
Positive integers can be compared in the usual way.
Compare:
4 and 9
On the number line, 9 lies farther to the right.
Therefore:
9 > 4
or:
4 < 9
Comparing a Positive and a Negative Integer
Compare:
5 and −3
5 is to the right of zero.
−3 is to the left of zero.
Therefore:
5 > −3
This leads to an important rule:
Every positive integer is greater than every negative integer.
Why Are Negative Numbers Less Than Positive Numbers?
Positive integers lie to the right of zero.
Negative integers lie to the left of zero.
Since values increase as we move right:
negative integer < 0 < positive integer
For example:
−4 < 0 < 6
Therefore:
−4 < 6
Even a Small Positive Number Is Greater
Compare:
1 and −100
The digits in 100 are much larger than the digit in 1.
However, this does not determine the comparison.
−100 is far to the left of zero.
1 is to the right of zero.
Therefore:
1 > −100
The signs matter.
Comparing Negative Integers
Comparing two negative integers requires careful thinking.
Compare:
−3 and −8
On the number line:
−8 is farther left.
−3 is farther right.
Therefore:
−3 > −8
For negative integers:
The number closer to zero is greater.
Why Is −3 Greater Than −8?
Think about temperature.
−3°C
is warmer than:
−8°C
Therefore:
−3 > −8
Or think about money.
A balance of:
−$3
represents a smaller debt than:
−$8
So −3 represents the greater numerical value.
Another Negative Comparison
Compare:
−15 and −6
−6 is closer to zero.
Therefore:
−6 > −15
We can also write:
−15 < −6
Do not simply compare 15 and 6 while ignoring the negative signs.
Comparing with Zero
Zero is greater than every negative integer.
For example:
0 > −5
0 > −100
0 > −1
Zero is less than every positive integer.
For example:
0 < 2
0 < 18
0 < 1,000
Therefore:
negative integers < 0 < positive integers
A Reliable Comparison Strategy
When comparing two integers:
Step 1: Imagine or draw a number line.
Step 2: Locate both integers.
Step 3: Determine which integer is farther to the right.
Step 4: The integer farther right is greater.
This method works for every pair of integers.
Worked Example 1
Compare:
−7 and 4
−7 is left of zero.
4 is right of zero.
Therefore:
−7 < 4
Worked Example 2
Compare:
−9 and −2
−2 is farther right.
Therefore:
−2 > −9
or:
−9 < −2
Worked Example 3
Compare:
0 and −14
Zero is farther right.
Therefore:
0 > −14
Worked Example 4
Compare:
12 and 0
12 is farther right.
Therefore:
12 > 0
Worked Example 5
Compare:
−25 and −30
−25 is closer to zero and farther right.
Therefore:
−25 > −30
Ordering Integers
To order integers means to arrange them according to their numerical value.
Integers can be ordered:
least to greatest
or:
greatest to least
A number line makes both processes easier.
Least to Greatest
Least to greatest means:
smallest → largest
On a number line, this means reading:
left → right
For example:
−5, −2, 0, 3, 7
is ordered from least to greatest.
We can write:
−5 < −2 < 0 < 3 < 7
Greatest to Least
Greatest to least means:
largest → smallest
On a number line, read:
right → left
For example:
7, 3, 0, −2, −5
We can write:
7 > 3 > 0 > −2 > −5
Ordering a Mixed Set of Integers
Order from least to greatest:
4, −6, 2, −1, 0
First identify the negative integers:
−6, −1
Then zero:
0
Then positive integers:
2, 4
Therefore:
−6 < −1 < 0 < 2 < 4
Ordering Several Negative Integers
Order from least to greatest:
−3, −12, −5, −1
Think about their positions on the number line.
The number farthest left is:
−12
Then:
−5
Then:
−3
Then:
−1
Therefore:
−12 < −5 < −3 < −1
Ordering from Greatest to Least
Order:
−8, 5, 0, −2, 7, −11
from greatest to least.
Positive integers first:
7, 5
Then:
0
Then negative integers from closest to zero to farthest:
−2, −8, −11
Therefore:
7 > 5 > 0 > −2 > −8 > −11
A Shortcut for Ordering Mixed Integers
For a set containing positive integers, zero, and negative integers:
For least to greatest:
- most negative values
- negative values closer to zero
- zero
- small positive values
- larger positive values
For greatest to least, reverse the order.
Absolute Value and Integer Comparisons
The absolute value of an integer is its distance from zero.
For example:
|−8| = 8
|5| = 5
Be careful:
A greater absolute value does not always mean a greater integer.
For example:
|−8| > |5|
but:
−8 < 5
Opposites and Comparisons
Opposite integers are the same distance from zero but on opposite sides.
Examples:
−4 and 4
−10 and 10
For any positive integer:
n > −n
For example:
8 > −8
because 8 lies to the right of −8.
Number-Line Distance Is Different from Value
Consider:
−12 and 5
−12 is farther from zero:
|−12| = 12
while:
|5| = 5
But:
−12 < 5
So we must distinguish between:
distance from zero
and:
numerical value
Integers in Temperature
Temperature provides a useful real-world example.
Suppose the temperatures are:
−8°C, 4°C, −2°C, 0°C, 6°C
From coldest to warmest:
−8°C < −2°C < 0°C < 4°C < 6°C
The coldest temperature has the smallest numerical value.
Temperature Problem
Four cities record:
City A: −12°C
City B: −3°C
City C: 5°C
City D: −7°C
Order from coldest to warmest:
−12°C < −7°C < −3°C < 5°C
Therefore:
- City A is coldest.
- City C is warmest.
Integers and Elevation
Elevation can be measured relative to sea level.
Suppose:
Location A: +350 m
Location B: −40 m
Location C: +75 m
Location D: −120 m
From lowest to highest:
−120 m < −40 m < 75 m < 350 m
Integers and Depth
Suppose three divers are located at:
−6 m
−15 m
−9 m
Which diver is deepest?
Order:
−15 < −9 < −6
Therefore:
−15 m
represents the deepest position.
The deepest location has the smallest numerical value.
Integers and Money
Integers can represent financial balances or changes.
For example:
+$80
can represent a gain of $80.
−$25
can represent a loss of $25.
$0
can represent no gain or loss.
Comparing Debts
Suppose:
Person A has a balance of:
−$20
Person B has a balance of:
−$75
Numerically:
−20 > −75
Person A's balance is greater because it is closer to zero.
Integer Changes
Suppose a business records:
+$200
−$150
+$75
−$300
$0
Order from least to greatest:
−$300 < −$150 < $0 < +$75 < +$200
The largest loss is represented by the smallest integer.
Integers and Building Floors
Suppose an elevator can stop at:
−3, −2, −1, 0, 1, 2, 3, 4
where:
0 = ground level
Negative floors are underground.
Positive floors are above ground.
Floor:
−3
is lower than:
−1
because:
−3 < −1
Integers and Direction
Suppose we define:
east = positive
west = negative
Then:
+8 km
means 8 km east of the starting point.
−5 km
means 5 km west.
Compare:
−5 < 8
The positions can be represented directly on a number line.
Integers and Sports
Some sports use positive and negative values.
In golf, a score relative to par might be:
Player A: −4
Player B: +2
Player C: −1
Player D: 0
Numerically, from least to greatest:
−4 < −1 < 0 < 2
Remember that in some real-world contexts, the numerically greatest value is not necessarily the "best" result. Interpretation depends on the situation.
Worked Example 6: Ordering Temperatures
Order from least to greatest:
3°C, −5°C, 0°C, −1°C, 7°C
Answer:
−5°C < −1°C < 0°C < 3°C < 7°C
Worked Example 7: Ordering Elevations
Order from highest to lowest:
−20 m, 150 m, 0 m, 85 m, −60 m
Answer:
150 m > 85 m > 0 m > −20 m > −60 m
Worked Example 8: Comparing Balances
Account A:
−$45
Account B:
$12
Since every positive number is greater than every negative number:
12 > −45
Therefore, Account B has the greater balance.
Worked Example 9: Comparing Two Negative Values
Compare:
−47 and −52
−47 is closer to zero.
Therefore:
−47 > −52
Worked Example 10: Mixed Integers
Order from least to greatest:
12, −4, −15, 8, 0, −1, 5
Answer:
−15 < −4 < −1 < 0 < 5 < 8 < 12
Finding Missing Integers
Integer comparisons can also be used to determine possible missing values.
Suppose:
−5 < x < 2
and x must be an integer.
Possible values include:
−4, −3, −2, −1, 0, 1
This means x must lie between −5 and 2 on the number line.
Another Missing-Integer Problem
Suppose:
−8 < x < −3
Possible integer values are:
−7, −6, −5, −4
Notice that:
−2
does not satisfy the condition because:
−2 > −3
Reasoning with Integer Comparisons
Consider the statement:
A < B
This means A lies to the left of B on a number line.
If:
A = −7
and:
B = −2
then:
−7 < −2
because −7 lies farther left.
Number-line reasoning helps us explain rather than simply memorize comparison rules.
Real-World Problem 1: Weather
Morning temperature:
−6°C
Afternoon temperature:
2°C
Night temperature:
−4°C
Order from coldest to warmest:
−6°C < −4°C < 2°C
Real-World Problem 2: Ocean Depth
Three objects are located at:
Object A: −35 m
Object B: −12 m
Object C: −48 m
From deepest to shallowest:
−48 m, −35 m, −12 m
Object C is deepest because −48 is the smallest integer.
Real-World Problem 3: Building Floors
Three people are on floors:
A: −2
B: 5
C: −4
Order from lowest to highest:
−4 < −2 < 5
Person C is on the lowest floor.
Real-World Problem 4: Financial Changes
A company records:
Monday: +$120
Tuesday: −$80
Wednesday: −$150
Thursday: +$45
Order the changes from least to greatest:
−$150 < −$80 < +$45 < +$120
Real-World Problem 5: Temperature Records
Five temperatures are:
−2°C, −11°C, 4°C, −6°C, 1°C
The minimum temperature is:
−11°C
The maximum temperature is:
4°C
The values in order are:
−11 < −6 < −2 < 1 < 4
Comparing Integers Without Drawing a Number Line
Once the number-line idea is understood, comparisons can often be made mentally.
Ask:
Are the signs different?
If yes, the positive integer is greater.
Are both positive?
The larger magnitude is greater.
Are both negative?
The integer closer to zero is greater.
Is one number zero?
Zero is greater than any negative integer and less than any positive integer.
Common Mistakes
Mistake 1: Ignoring negative signs
Incorrect:
−10 > −3
because 10 > 3.
Correct:
−10 < −3
Mistake 2: Thinking the negative number with larger digits is greater
Incorrect:
−50 > −8
Correct:
−50 < −8
because −50 lies farther left.
Mistake 3: Thinking zero is negative
Zero is neither positive nor negative.
Mistake 4: Ordering negative integers like positive integers
Incorrect least-to-greatest order:
−2, −5, −9
Correct:
−9, −5, −2
Mistake 5: Confusing value with absolute value
Although:
|−20| > |5|
we still have:
−20 < 5
Error Analysis
A student orders:
−3, −8, −12
from least to greatest.
This is incorrect.
Imagine the number line:
−12 is farthest left.
Then:
−8
Then:
−3
Correct order:
−12 < −8 < −3
Another Error Analysis
A student says:
−100 > 2
because 100 is greater than 2.
This ignores the signs.
−100 is negative.
2 is positive.
Every positive integer is greater than every negative integer.
Therefore:
−100 < 2
Explaining Your Reasoning
A strong mathematical explanation should say why a comparison is true.
Instead of only writing:
−4 > −9
you could write:
−4 is greater than −9 because −4 lies farther to the right on the number line.
Or:
−4 is closer to zero than −9, so −4 is the greater negative integer.
A Reliable Ordering Strategy
When ordering integers:
Step 1: Identify all negative integers.
Step 2: Identify zero, if present.
Step 3: Identify all positive integers.
Step 4: Imagine their positions on a number line.
Step 5: For least to greatest, read from left to right.
Step 6: For greatest to least, read from right to left.
Step 7: Check negative integers carefully.
Did You Know?
A number line has no beginning or end.
The integers continue forever in both directions:
..., −1002, −1001, −1000, ...
and:
..., 1000, 1001, 1002, ...
No matter how large a positive integer you choose, a greater integer exists.
No matter how negative an integer is, a smaller integer exists.
This is why the arrows on a number line extend in both directions.
Key Terms
- Integer: Positive whole number, negative whole number, or zero.
- Positive integer: Integer greater than zero.
- Negative integer: Integer less than zero.
- Zero: Integer that is neither positive nor negative.
- Compare: Determine which value is greater, smaller, or equal.
- Order: Arrange numbers according to value.
- Least: Smallest numerical value.
- Greatest: Largest numerical value.
- Number line: Visual representation of numbers according to position and value.
- Greater than: Larger in numerical value; represented by >.
- Less than: Smaller in numerical value; represented by <.
- Absolute value: Distance of an integer from zero.
- Opposite integers: Integers the same distance from zero on opposite sides.
- Minimum: Smallest value in a set.
- Maximum: Greatest value in a set.
Key Relationships
On a number line:
left = smaller
right = greater
For negative and positive integers:
negative < 0 < positive
For example:
−8 < 0 < 5
For two negative integers:
the number closer to zero is greater
For example:
−3 > −10
Opposite integers:
−a < a
when a is positive.
For example:
−7 < 7
Key Takeaways
- Integers can be compared using their positions on a number line.
- Numbers increase as you move to the right.
- Numbers decrease as you move to the left.
- The integer farther to the right is always greater.
- Every positive integer is greater than every negative integer.
- Zero is greater than every negative integer.
- Zero is less than every positive integer.
- When comparing two negative integers, the number closer to zero is greater.
- A large absolute value does not necessarily mean a large numerical value.
- Least to greatest means arranging integers from smallest to largest.
- On a number line, least to greatest corresponds to left to right.
- Greatest to least means arranging integers from largest to smallest.
- On a number line, greatest to least corresponds to right to left.
- Negative integers must be ordered carefully because integers farther from zero in the negative direction are smaller.
- Number-line reasoning is more reliable than comparing digits alone.
- Integer comparisons can be used with temperatures, elevations, depths, financial balances, building floors, directions, and changes.
- Real-world context can affect how an integer should be interpreted.
- Minimum means the smallest numerical value, while maximum means the greatest.
- Strong mathematical explanations use number-line position or distance from zero to justify comparisons.
- Comparing and ordering integers provides the foundation for later work with integer addition, subtraction, inequalities, coordinate geometry, algebra, and many real-world mathematical models.