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.

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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.

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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.

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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.

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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

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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.

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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

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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.

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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.

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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.

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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.

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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

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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.

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Worked Example 1

Compare:

−7 and 4

−7 is left of zero.

4 is right of zero.

Therefore:

−7 < 4

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Worked Example 2

Compare:

−9 and −2

−2 is farther right.

Therefore:

−2 > −9

or:

−9 < −2

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Worked Example 3

Compare:

0 and −14

Zero is farther right.

Therefore:

0 > −14

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Worked Example 4

Compare:

12 and 0

12 is farther right.

Therefore:

12 > 0

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Worked Example 5

Compare:

−25 and −30

−25 is closer to zero and farther right.

Therefore:

−25 > −30

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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.

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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

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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

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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

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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

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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

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A Shortcut for Ordering Mixed Integers

For a set containing positive integers, zero, and negative integers:

For least to greatest:

1. most negative values
2. negative values closer to zero
3. zero
4. small positive values
5. larger positive values

For greatest to least, reverse the order.

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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

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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.

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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

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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.

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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.

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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

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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.

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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.

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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.

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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.

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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

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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.

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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.

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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

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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

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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.

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Worked Example 9: Comparing Two Negative Values

Compare:

−47 and −52

−47 is closer to zero.

Therefore:

−47 > −52

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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

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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.

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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

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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.

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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

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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.

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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.

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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

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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

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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.

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Common Mistakes

Mistake 1: Ignoring negative signs

Incorrect:

−10 > −3

because 10 > 3.

Correct:

−10 < −3

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Mistake 2: Thinking the negative number with larger digits is greater

Incorrect:

−50 > −8

Correct:

−50 < −8

because −50 lies farther left.

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Mistake 3: Thinking zero is negative

Zero is neither positive nor negative.

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Mistake 4: Ordering negative integers like positive integers

Incorrect least-to-greatest order:

−2, −5, −9

Correct:

−9, −5, −2

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Mistake 5: Confusing value with absolute value

Although:

|−20| > |5|

we still have:

−20 < 5

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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

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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

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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.

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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.

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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.

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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.

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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

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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.