Integers

What are Integers?

Integers are whole numbers that can be positive, negative, or zero: …-3, -2, -1, 0, 1, 2, 3…

They do not include fractions or decimals. The set of integers is often written with the symbol ℤ (from the German word Zahlen, meaning numbers).

Key Terms

• Integer — any whole number: negative, zero, or positive
• Positive integer — a whole number greater than zero (1, 2, 3…)
• Negative integer — a whole number less than zero (-1, -2, -3…)
• Absolute value — the distance from zero, always non-negative (|-5| = 5)

The Number Line

←  -5  -4  -3  -2  -1  0  1  2  3  4  5  →

Numbers to the left are smaller. Numbers to the right are larger. This means:

• -5 < -1 < 0 < 3 < 7
• -100 < -1 (even though 100 looks bigger, the negative sign makes it much smaller)

Adding Integers

There are three cases when adding integers:

• Positive + Positive: add normally — 4 + 3 = 7
• Negative + Negative: add the sizes, keep the negative sign — (-3) + (-5) = -8
• Mixed signs: find the difference of the sizes, use the sign of the larger — 7 + (-4) = 3, and (-9) + 4 = -5

Worked Example: What is (-6) + 10? The sizes are 6 and 10. The difference is 4. The larger number (10) is positive, so the answer is +4.

Subtracting Integers

The key rule: subtracting a negative is the same as adding a positive.

a - (-b) = a + b

• 5 - (-3) = 5 + 3 = 8
• -2 - (-7) = -2 + 7 = 5
• 4 - (-4) = 4 + 4 = 8

Think of it as: two negatives cancel each other out and become a positive.

Multiplying Integers

Signs	Result	Example
Positive × Positive	Positive	3 × 4 = 12
Negative × Negative	Positive	(-3) × (-4) = 12
Positive × Negative	Negative	3 × (-4) = -12
Negative × Positive	Negative	(-3) × 4 = -12

Key rule: Same signs → positive. Different signs → negative.

Dividing Integers

The same sign rules that apply to multiplication also apply to division:

• (-8) ÷ (-2) = 4 (same signs → positive)
• 15 ÷ (-3) = -5 (different signs → negative)
• (-20) ÷ 4 = -5 (different signs → negative)

Worked Example: (-8) ÷ (-2) — both signs are negative (same signs) → result is positive → 4

Powers of Negative Numbers

Be careful with negative bases raised to a power:

• (-6)² = (-6) × (-6) = 36 (negative × negative = positive)
• (-2)³ = (-2) × (-2) × (-2) = 4 × (-2) = -8 (odd power stays negative)

Even powers of negative numbers are always positive. Odd powers are always negative.

Absolute Value

The absolute value of an integer is its distance from zero on the number line — always non-negative.

• |7| = 7
• |-7| = 7
• |0| = 0

Absolute value is written with vertical bars: |n|. It answers: "how far from zero?" not "in which direction?"

Ordering and Comparing Integers

To order integers, place them on a number line or compare their positions:

• More negative = further left = smaller value
• Example: put -3, 1, -7, 0, 5 in ascending order: -7, -3, 0, 1, 5

Real-World Context

Negative integers appear everywhere in daily life:

• Temperature: -10°C is 10 degrees below freezing
• Altitude: -400 m means 400 metres below sea level
• Finance: -£50 means you owe £50 (a debt)
• Time: -3 years means 3 years before a reference point (e.g., 3 BC)
• Sport scores: a golf score of -4 is 4 under par

Common Mistakes

• Mistake: thinking -10 > -2 because 10 > 2. Fix: on the number line, -10 is further left, so -10 < -2.
• Mistake: (-3)² = -9. Fix: (-3)² = (-3) × (-3) = +9. The exponent applies to the whole negative term.
• Mistake: -5 - 3 = -2. Fix: -5 - 3 = -5 + (-3) = -8 (both negatives, add the sizes).
• Mistake: thinking a - (-b) stays as a - b. Fix: subtracting a negative always becomes addition: a + b.

Tips and Tricks

• Draw a number line whenever you are unsure — it makes direction visual.
• For multiplication and division, work out the size first, then apply the sign rule separately.
• The product of an even number of negatives is positive; an odd number of negatives is negative.
• To check a subtraction answer, use the inverse: if 8 - (-3) = 11, then 11 + (-3) should equal 8. ✓