Integers — Class 6 IMO

Negative Numbers, Number Line & Ordering

What are Integers?

Integers are whole numbers and their opposites. They include:

Positive integers: 1, 2, 3, 4... (to the right of zero)
Zero: 0 (neither positive nor negative)
Negative integers: -1, -2, -3, -4... (to the left of zero)

The Number Line:

<----|----|----|----|----|----|----|----|----|----|---->
    -5   -4   -3   -2   -1    0    1    2    3    4    5
    
Negative ←                      → Positive

Absolute Value:

Distance from zero (always positive)
Written as |−5| = 5
|3| = 3

INTEGER NUMBER LINE - ANNOTATED:

 Negative Numbers Positive Numbers
 ↓ ↓
 <----●----●----●----●----●----●----●----●----●----●---->
 -5 -4 -3 -2 -1 0 1 2 3 4 5
 
 ←── Getting Smaller ──┼── Getting Larger ──→


PLACING INTEGERS ON NUMBER LINE:

 Where is -3?
 
 <----|----|----|----|----|----|----|---->
 -4 -3 -2 -1 0 1 2
 ↑
 -3 is here!


ABSOLUTE VALUE VISUAL:

 | -4 | = 4 | 4 | = 4
 
 <----|----|----|----|----|----|----|---->
 -4 -3 -2 -1 0 1 2 3 4
 ↑ ↑
 Distance = 4 Distance = 4


REAL-LIFE EXAMPLES OF NEGATIVES:

 Temperature: -5°C (below freezing)
 Elevation: -100 m (below sea level)
 Money: -20 (debt/owed)
 Football: -7 yards (loss)
 Time: -3 hours (3 hours ago)

Example 1: Which is greater, −3 or −7?

−3 is to the right of −7, so −3 is greater.

Example 2: Arrange in ascending order: −2, 3, −5, 0.

−5, −2, 0, 3.

Example 3:

Draw a number line from -5 to 5 and place -2, 0, 3, -4.

``` <----|----|----|----|----|----|----|----|----|----|---->

-5 -4 -3 -2 -1 0 1 2 3 4 5 ↑ ↑ ↑ ↑

-4 -2 0 3 ```

Example 4:

Which is greater: -5 or -8?

On a number line, numbers increase to the right
-5 is to the right of -8
Therefore -5 > -8
Answer: -5

Example 5:

Find |−7| and |3|

|−7| = 7 (distance from 0 is 7 units)
|3| = 3 (distance from 0 is 3 units)
Answer: 7 and 3

Quick recap

Integers include negatives, zero and positives.
Further right on the number line means greater.
Integers = ..., -3, -2, -1, 0, 1, 2, 3, ...
Left of zero = negative, Right of zero = positive
Larger numbers are to the RIGHT
Absolute value = distance from zero (always positive)

✓ Quick check

Which integer is the smallest?

−4 is furthest left, so it is the smallest.

The opposite of +6 is ___ ?

The opposite of +6 is −6.

Addition & Subtraction of Integers

Adding Integers Using Number Line:

Adding a positive = move RIGHT
Adding a negative = move LEFT

Rules for Adding Integers:

First Number	Operation	Second Number	Result
Positive	+	Positive	Positive (add)
Positive	+	Negative	Subtract, sign of larger
Negative	+	Positive	Subtract, sign of larger
Negative	+	Negative	Negative (add absolute values)

Subtracting Integers:

Rule: Subtracting a number = adding its opposite

a - b = a + (-b)
a - (-b) = a + b

Examples:

5 - 3 = 5 + (-3) = 2
5 - (-3) = 5 + 3 = 8
-5 - 3 = -5 + (-3) = -8
-5 - (-3) = -5 + 3 = -2

ADDING INTEGERS ON NUMBER LINE:

    Example 1: 3 + (-5) = -2
    
    Start at 3, move LEFT 5 spaces
    
    <----|----|----|----|----|----|----|---->
        -3   -2   -1    0    1    2    3    4
              ↑    ↑    ↑    ↑    ↑
        End  -2   -1    0    1    2   Start 3
        (←5 steps)


    Example 2: -2 + 4 = 2
    
    Start at -2, move RIGHT 4 spaces
    
    <----|----|----|----|----|----|----|---->
        -3   -2   -1    0    1    2    3    4
              ↑    ↑    ↑    ↑
            Start  -1    0    1   End 2
            (-2)   (←4 steps →)


SUBTRACTION AS ADDING OPPOSITE:

    Subtraction problem:  5 - 8 = ?
    
    Step 1: Change to addition: 5 + (-8)
    Step 2: Start at 5, move left 8: 5 → 4 → 3 → 2 → 1 → 0 → -1 → -2 → -3
    Answer: -3
    
    
    Subtraction problem: -3 - (-7) = ?
    
    Step 1: Change to addition: -3 + 7
    Step 2: Start at -3, move right 7: -3 → -2 → -1 → 0 → 1 → 2 → 3 → 4
    Answer: 4

Example 1: Add −5 + 3.

Different signs: 5 − 3 = 2, keep the sign of 5, so −2.

Example 2: Subtract −4 − 6.

−4 − 6 = −10.

Example 3:

Add: 5 + (-3)

Start at 5, move left 3
5 → 4 → 3 → 2
Answer: 2

Example 4:

Add: -4 + (-2)

Start at -4, move left 2
-4 → -5 → -6
Answer: -6

Example 5:

Subtract: 8 - 12

8 - 12 = 8 + (-12)
Start at 8, move left 12
8 → 7 → 6 → 5 → 4 → 3 → 2 → 1 → 0 → -1 → -2 → -3 → -4
Answer: -4

Example 6:

Subtract: -6 - (-4)

-6 - (-4) = -6 + 4
Start at -6, move right 4
-6 → -5 → -4 → -3 → -2
Answer: -2

Example 7:

Simplify: 3 + (-7) + 2

First: 3 + (-7) = -4
Then: -4 + 2 = -2
Answer: -2

Quick recap

Same sign: add and keep the sign.
Different signs: subtract and keep the larger number's sign.
Adding positive = move right
Adding negative = move left
Subtraction = add the opposite (keep-change-change)
Same signs: add and keep sign
Different signs: subtract and keep sign of larger absolute value
Practice with small numbers first

✓ Quick check

What is 7 + (−10)?

10 − 7 = 3, keep the negative sign, so −3.

What is −8 + 8?

Opposites add to 0.

Multiplication, Division & Word Problems

When multiplying or dividing integers, like signs give a positive answer and unlike signs give a negative answer.

So (−4) × 3 = −12, but (−12) ÷ (−4) = 3. These rules help with temperature, height and profit/loss problems.

Example 1: Multiply (−4) × 3.

Unlike signs give a negative: −12.

Example 2: Divide (−12) ÷ (−4).

Like signs give a positive: 3.

Quick recap

Like signs → positive; unlike signs → negative.
Useful for temperature, height and profit/loss.

✓ Quick check

What is (−5) × (−6)?

Like signs give a positive: 30.

What is (−20) ÷ 5?

Unlike signs give a negative: −4.