Practice Maths

L45 — Comparing and Ordering Integers

Key Ideas

Key Terms

integer
A whole number (positive, negative, or zero) with no fractional or decimal part. Examples: … −3, −2, −1, 0, 1, 2, 3, …
positive integer
A whole number greater than zero: 1, 2, 3, …
negative integer
A whole number less than zero: … −3, −2, −1.
number line
A visual representation of numbers arranged in order from left (smallest) to right (largest). Moving right increases value; moving left decreases it.
absolute value
The distance of a number from zero on the number line. Always positive or zero. Written as |n|. For example, |−7| = 7 and |7| = 7.

Integers are whole numbers including negatives: … −3, −2, −1, 0, 1, 2, 3, …

Number line: integers further to the right are larger. Zero sits in the middle.

Absolute value |n| is the distance from zero, so it is always positive or zero. |−7| = 7 and |7| = 7.

Ordering: the most negative integer is the smallest. Arrange from left to right on the number line.

Hot Tip: −100 < −1 even though 100 > 1. On the number line, −100 is much further to the left, so it is smaller.

Worked Example 1 — Order −5, 3, −2, 0, 7, −8 from smallest to largest

Place on number line: −8 is leftmost, then −5, −2, 0, 3, 7.

Answer: −8, −5, −2, 0, 3, 7

Worked Example 2 — Absolute value

|−7| = 7    |3| = 3    |0| = 0

Compare |−7| and |3|: 7 > 3, so −7 is further from zero than 3.

The Number Line: Your Best Friend

The number line extends infinitely in both directions. Integers to the RIGHT are always larger, and integers to the LEFT are always smaller. This simple visual makes comparing any two integers straightforward — just find which is further right.

…−5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5…

−3 is to the right of −5, so −3 > −5. This surprises many students because 5 > 3 for positive numbers — but flip the sign and the order reverses. The more negative a number, the smaller it is.

Temperature and Real-World Meaning

Antarctica's winter temperatures sometimes reach −70°C. That's much colder than −10°C (a cold Sydney winter morning). In terms of integers: −70 < −10. The coldest temperatures are the most negative, which means they're the smallest numbers.

Golf uses negative integers to represent shots under par. A score of −4 is better than −2 in golf, but −4 < −2 mathematically. Context matters! In golf, lower score = better. In maths, higher value = bigger. Always read the context carefully.

Remember: On the number line, further RIGHT = larger. −100 < −1, even though 100 > 1. The sign flips the order for negative numbers. When in doubt, draw a small number line and place both numbers on it.

Absolute Value: Distance from Zero

The absolute value |n| of a number is its distance from zero on the number line — always positive (or zero). |−7| = 7 because −7 is 7 units away from zero. |+7| = 7 for the same reason.

Absolute value is useful when we care about the SIZE of a number but not its direction. For example, if you're 3 km north OR 3 km south of home, you've walked the same distance — |3| = |−3| = 3 km.

Ordering a Mixed Set

To order integers like −5, 3, −2, 0, 7, −8 from smallest to largest:

  1. Place them mentally on a number line from left to right.
  2. Negatives come first (most negative is smallest): −8, −5, −2
  3. Then zero: 0
  4. Then positives (smallest first): 3, 7
  5. Answer: −8, −5, −2, 0, 3, 7
Common Mistake: Ordering negative numbers as if they were positive (e.g., writing −8, −5, −2 as "largest to smallest" negative, then reversing). Instead, think of temperature: −8°C is colder (smaller) than −5°C. The most negative is always the smallest.

  1. Find the absolute value

    1. |−5|
    2. |3|
    3. |−12|
    4. |0|
    5. |−7|
    6. |15|
    7. |−3|
    8. |8|

  2. Compare each pair using <, > or =

    Write <, > or = in each box.

    1. −3 −7
    2. −1 0
    3. 5 −5
    4. −10 −4
    5. 0 −2
    6. −8 −9
    7. −15 −15
    8. 6 −100

  3. Order each set from smallest to largest

    1. −3, 1, −5, 0, 4
    2. 7, −2, −7, 3, −1
    3. −10, −4, −8, 0, −1
    4. 6, −6, 2, −2, 0
    5. −15, 5, −5, 10, −10, 0
    6. −100, −1, −50, −10, −25
    7. 4, −9, 0, −4, 9, −1
    8. −3, −30, −3, 3, 30

  4. Temperature and depths

    1. The temperature is −8°C on Monday and −3°C on Tuesday. Which day is colder?
    2. A fish is at −12 m depth. A submarine is at −80 m. Which is deeper?
    3. True or false: −20 > −15. Explain.
    4. True or false: the absolute value of a negative number is always positive. Give an example.

  5. Number line reasoning

    1. Which of these is further from zero: −9 or 7?
    2. List all integers between −5 and −1 (not including −5 and −1).
    3. What integer is exactly halfway between −6 and 4?
    4. Two numbers have absolute value 6. What are they?

  6. Real-world contexts

    1. Five cities recorded these overnight temperatures: −12°C, 3°C, −5°C, 0°C, −9°C. Rank them from coldest to warmest.
    2. In golf, lower scores are better. Players scored −3, +2, −1, +4, 0, −5. List them from best (lowest) to worst.
    3. Bank balances: Alice: $−45, Bob: $−120, Carla: $30, Dave: $−15. Who has the lowest balance? Who is best off?
    4. A diver is at −25 m. She rises 10 m. What is her new depth? Is she above or below −20 m?

  7. Order from largest to smallest

    1. 8, −3, 0, −11, 5
    2. −2, −9, 1, −4, 6
    3. 0, −7, −7, 7, −1
    4. −50, 20, −20, 50, 0
    5. 3, −3, −33, 33, −13, 13
    6. −200, 150, −150, 200, −100

  8. True or false — justify each answer

    1. |−4| > |3|
    2. −7 > −6
    3. |−10| = |10|
    4. The largest negative integer is −1.
    5. 0 is a positive integer.
    6. |−3| < |−5|

  9. Absolute value comparisons

    Write = or ≠ in each box.

    1. |−9| |9|
    2. |−4| |−6|
    3. |0| |−1|
    4. |−15| |12|
    5. |−8| |8|
    6. |−100| |−99|
    7. |2| |−2|
    8. |−30| |25|

  10. Integer puzzles

    1. I am an integer. My absolute value is 8. I am less than 0. What am I?
    2. I am an integer between −10 and −6 (not including either). List all possible values.
    3. Two integers have a sum of 0 and are not equal to each other. One integer is 13. What is the other?
    4. A football team's score changes are: −7, +3, −4, +6, −2. List the changes in order from least to greatest change.
    5. The temperature at midnight was −6°C. It fell further to −14°C by 3 am. By how many degrees did it drop? Write the answer as a positive number.
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