Positive and Negative Integers- Operations and Rules

What Integers Actually Are

Integers are whole numbers. No decimals, no fractions, no nonsense. They include positive numbers, negative numbers, and zero. That's it. If you see a decimal point, you're not dealing with integers anymore.

The number line is your visual anchor here. Picture a horizontal line with zero sitting right in the middle. Numbers to the right are positive. Numbers to the left are negative. Every integer has a specific spot.

This isn't abstract math theory. It's the foundation for everything from your bank account to temperature readings to elevator floors.

Positive vs. Negative Integers: The Core Difference

Positive integers are greater than zero. They represent quantities, gains, temperatures above freezing, floors above ground level. They don't have a sign in front of them, or they have a plus sign (+).

Negative integers are less than zero. They represent debts, losses, temperatures below freezing, floors below ground level. They always have a minus sign (−) in front of them.

Zero is neither positive nor negative. It's the dividing line. Some people struggle with this, but zero just sits there being neutral.

The Four Operations: How They Actually Work

Addition, subtraction, multiplication, division. Each has specific rules when negative numbers enter the picture. Memorize these rules or you'll constantly second-guess yourself.

Addition Rules

Adding positive to positive: just add them. Easy.

Adding negative to negative: add the absolute values, keep the negative sign. Example: −3 + (−5) = −8.

Adding positive to negative: subtract the smaller absolute value from the larger one. The result takes the sign of the number with the larger absolute value. Example: −7 + 4 = −3. Example: −3 + 8 = 5.

Subtraction Rules

Here's where people get messy. Subtracting a negative is the same as adding a positive. This is the rule everyone forgets.

So: 5 − (−3) = 5 + 3 = 8

Keep the first number, change the subtraction to addition, flip the sign of the second number. That's the process.

For positive minus negative: 5 − (−3) = 8

For negative minus positive: −5 − 3 = −8 (just keep subtracting)

For negative minus negative: −5 − (−3) = −5 + 3 = −2

Multiplication Rules

Two rules to remember:

• Positive × Positive = Positive
• Negative × Negative = Positive
• Positive × Negative = Negative
• Negative × Positive = Negative

Same signs give you a positive. Different signs give you a negative. Multiply the absolute values, then apply the sign rule.

Example: (−4) × (−6) = 24

Example: (−4) × 6 = −24

Division Rules

Division follows the exact same sign rules as multiplication.

• Same signs = Positive result
• Different signs = Negative result

Example: (−20) ÷ (−5) = 4

Example: (−20) ÷ 5 = −4

Example: 20 ÷ (−5) = −4

Signs Cheat Sheet

Double Negatives: The Trap Everyone Falls Into

−(−5) = 5. Two negatives cancel out. This applies to multiplication and when you see parentheses with a negative sign in front.

But don't confuse this with subtraction. In subtraction, you convert the minus to plus and flip the sign of the number being subtracted.

These are similar concepts but different operations. Keep them separate in your head.

Getting Started: Practice Problems

Work through these without a calculator first. Check your answers.

1. (−12) + 7 = ?

2. 15 − (−8) = ?

3. (−3) × (−9) = ?

4. (−36) ÷ 6 = ?

5. (−4) + (−11) = ?

6. 5 − 12 = ?

7. (−2) × 7 = ?

8. (−24) ÷ (−3) = ?

Answers

1. −5    2. 23    3. 27    4. −6    5. −15    6. −7    7. −14    8. 8

If you missed any, go back and identify which rule applies. That's the only way to stop making the same mistake.

Common Mistakes to Avoid

• Forgetting that subtracting a negative flips to addition
• Losing track of the sign during multi-step problems
• Treating subtraction and addition of negatives as interchangeable operations
• Rushing through problems without checking sign rules

Slow down. Apply the rules systematically. Most errors come from skipping steps, not from misunderstanding the math.

Why This Matters

You encounter integers constantly. Bank balances go negative. Temperatures drop below zero. Elevation can be below sea level. If you can't work with negative numbers, you'll miscalculate real situations.

These rules aren't academic exercises. They're practical tools. Once you internalize them, working with integers becomes automatic.