Positive and Negative Integers — Definition, Formula & Examples
Positive and negative integers are whole numbers that sit on opposite sides of zero on the number line. Positive integers (1, 2, 3, …) are greater than zero, while negative integers (−1, −2, −3, …) are less than zero.
The set of integers, denoted , consists of all whole numbers: . This set is partitioned into three disjoint subsets: the positive integers , the negative integers , and the singleton . Zero is an integer but is neither positive nor negative.
How It Works
Picture a number line with zero in the center. Every step to the right is a positive integer, and every step to the left is a negative integer. To compare two integers, the one farther right is always greater — so even though 5 "looks bigger" than 3. When you add a positive integer, you move right; when you add a negative integer, you move left. Subtracting a negative integer is the same as adding its positive counterpart, because the two negatives cancel out. The sign of a product or quotient follows a simple rule: same signs give a positive result, and different signs give a negative result.
Worked Example
Problem: Compute −8 + 5.
Step 1: Start at −8 on the number line.
Step 2: Adding 5 means moving 5 units to the right.
Step 3: Since the signs are different, subtract the smaller absolute value from the larger: .
Step 4: The number with the larger absolute value is −8, which is negative, so the result keeps the negative sign.
Answer:
Another Example
This example focuses on multiplication of two negative integers, illustrating the sign rule, whereas the first example covers addition of integers with different signs.
Problem: Compute (−4) × (−6).
Step 1: Find the absolute values of each factor.
Step 2: Multiply the absolute values.
Step 3: Determine the sign. Both factors are negative, so the signs are the same. Same signs produce a positive product.
Answer:
Visualization
Why It Matters
Understanding positive and negative integers is essential in pre-algebra and algebra courses, where you constantly solve equations involving signed numbers. In real life, bank accounts use negative numbers for withdrawals and debt, thermometers show temperatures below zero, and scientists record elevations below sea level as negative values. Careers in finance, engineering, and data science all rely on fluent arithmetic with signed numbers.
Common Mistakes
Mistake: Thinking zero is a positive integer.
Correction: Zero is neither positive nor negative. It is the dividing point between the two groups. Positive integers start at 1.
Mistake: Assuming a number with a bigger digit is always greater (e.g., thinking ).
Correction: For negative integers, a larger absolute value means a smaller number. On the number line, is to the left of , so .
Mistake: Writing instead of .
Correction: Subtracting a negative is adding: . Remember to flip the subtraction of a negative into addition.
Check Your Understanding
Is greater than or less than ?
Hint: With negative numbers, a larger absolute value means a smaller actual value.
Answer: , because is farther to the left on the number line.
What is ?
Hint: Subtracting a negative is the same as adding a positive.
Answer: .
What is the sign of ?
Hint: Count the negative signs. An even number of negatives gives a positive product.
Answer: Positive. There are two negative factors (an even count), so the product is positive: .