Number Systems and Integer Operations
Master the full integer number line — positive, negative, zero — and learn to add, subtract, multiply, and divide integers with confidence and speed.
Learning Objectives
- 1Classify numbers as natural, whole, or integer and explain the relationships among these sets
- 2Compare and order integers using inequality symbols and the number line
- 3Compute absolute value and explain its geometric meaning as distance from zero
- 4Add, subtract, multiply, and divide integers fluently using sign rules
- 5Apply integer operations to real-world contexts including temperature, elevation, and finance
When Zero Is Not Enough
Imagine you are tracking the performance of a stock portfolio over five days. Monday: up $12. Tuesday: down $8. Wednesday: down $15. Thursday: up $3. Friday: down $6. What is the net change?
You cannot answer that with counting numbers alone. You need a system that handles direction — gain versus loss, above versus below, credit versus debt. That system is the integers: the positive whole numbers, the negative whole numbers, and zero.
This unit moves fast. You already know what negative numbers are. The goal here is to operate on them — fluently, accurately, and with genuine understanding of why the sign rules work.
The Number Sets
Mathematics builds number systems in layers:
- Natural numbers (N): 1, 2, 3, 4, ... (counting numbers)
- Whole numbers (W): 0, 1, 2, 3, 4, ... (natural numbers plus zero)
- Integers (Z): ... -3, -2, -1, 0, 1, 2, 3 ... (whole numbers plus negatives)
Every natural number is a whole number. Every whole number is an integer. The sets nest inside each other like boxes within boxes.
Think About
Why did mathematicians need to invent zero before they could invent negative numbers? What role does zero play in giving negative numbers their meaning?
Absolute Value: Distance Without Direction
The absolute value of an integer is its distance from zero on the number line. Distance is always non-negative.
|7| = 7 (seven units from zero)
|-7| = 7 (also seven units from zero)
|0| = 0
The numbers 7 and -7 are opposites — same distance from zero, opposite sides. Every integer has an opposite, and adding any integer to its opposite yields zero: 7 + (-7) = 0.
Precision matters: Absolute value is not "remove the negative sign." That shortcut works for single numbers but breaks for expressions. |3 - 10| = |-7| = 7, not 3 - 10 = -7. Always evaluate the expression inside the bars first, then take the distance.
Comparing and Ordering Integers
On the number line, right is greater, left is less. This means:
- -3 < 2 (negative numbers are always less than positive numbers)
- -1 > -5 (closer to zero means greater for negatives)
- -100 < -1 (farther left means less, regardless of the digit size)
Order these from least to greatest: 4, -7, 0, -2, 8, -7
Place them on the number line mentally: -7, -7, -2, 0, 4, 8.
❓Concept Check
Without a number line, explain why -999 is greater than -1,000. Use the concept of distance from zero in your reasoning.
▸
Concept Check
Without a number line, explain why -999 is greater than -1,000. Use the concept of distance from zero in your reasoning.
-999 is one unit closer to zero than -1,000. On the number line, -999 sits to the right of -1,000. Since right means greater, -999 > -1,000. The 'size' of the digit is irrelevant — what matters is position relative to zero.
Adding Integers
Same Signs: Add and Keep the Sign
When both numbers pull in the same direction, the result moves farther in that direction.
Different Signs: Subtract and Keep the Sign of the Larger Absolute Value
When the numbers pull in opposite directions, the stronger one wins. The result's sign matches the number with the larger absolute value.
Think About
A football team gains 8 yards, loses 12 yards, gains 3 yards, and loses 1 yard. Write this as an integer addition problem and find the net yardage. Does the order of the plays matter for the final result? Why or why not?
Subtracting Integers
Here is the single most important rule in this unit:
Subtracting an integer is the same as adding its opposite.
a - b = a + (-b)
This transforms every subtraction problem into an addition problem, which you already know how to solve.
That last one trips people up. Subtracting a negative number means adding a positive. Think of it as removing a debt — if someone cancels your $8 debt, you are $8 richer.
❓Concept Check
Calculate: -15 - (-22) + 7. Show each step by converting subtraction to addition.
▸
Concept Check
Calculate: -15 - (-22) + 7. Show each step by converting subtraction to addition.
-15 - (-22) + 7 = -15 + 22 + 7.
First: -15 + 22 = 7 (|22| > |-15|, result positive).
Then: 7 + 7 = 14.
Final answer: 14.
Practice: Khan Academy's integer operations exercises give immediate feedback. Work through at least 10 problems mixing addition and subtraction before moving on. Find them at 6th grade math (TX).
Multiplying Integers
The sign rules for multiplication follow a clean pattern:
| Factors | Result |
|---|---|
| positive x positive | positive |
| negative x negative | positive |
| positive x negative | negative |
| negative x positive | negative |
Same signs produce a positive product. Different signs produce a negative product.
Why Is Negative Times Negative Positive?
This is not an arbitrary rule. Consider the pattern:
Each time the first factor decreases by 1, the product increases by 2. Following the pattern: (-1) x (-2) = 2. The pattern demands it.
Think About
Extend the pattern: (-2) x (-2) = ? and (-3) x (-2) = ?. Does the pattern hold? Can you explain in words why multiplying two negatives must give a positive to keep arithmetic consistent?
Dividing Integers
Division follows the same sign rules as multiplication:
| Dividend / Divisor | Result |
|---|---|
| positive / positive | positive |
| negative / negative | positive |
| positive / negative | negative |
| negative / positive | negative |
Division by zero remains undefined. No integer multiplied by zero gives a nonzero result, so you cannot reverse the operation.
❓Concept Check
Compute: (-48) / 8 x (-2). Be careful with order of operations — multiplication and division happen left to right.
▸
Concept Check
Compute: (-48) / 8 x (-2). Be careful with order of operations — multiplication and division happen left to right.
(-48) / 8 = -6.
Then (-6) x (-2) = 12.
Final answer: 12.
The Stock Portfolio Revisited
Return to the opening problem: +12, -8, -15, +3, -6.
Net change = 12 + (-8) + (-15) + 3 + (-6)
Group the positives: 12 + 3 = 15
Group the negatives: (-8) + (-15) + (-6) = -29
Combine: 15 + (-29) = -14
The portfolio lost $14 over the week. Integer arithmetic made a messy real-world problem into a clean calculation.
Cross-Curricular Connection — Financial Markets: Gains and losses in stock portfolios are integer arithmetic at scale. Professional traders track daily P&L (profit and loss) using exactly the reasoning you just applied. Explore how markets aggregate millions of these calculations in First Securities Markets.
Cross-Curricular Connection — Math 6: The standard Math 6 course covers integer basics over more days with more scaffolding. If you want additional practice problems or a slower-paced explanation, see Integers.
Putting It All Together
Evaluate: -3 + 5 x (-2) - (-4)
Order of operations: multiplication first, then left to right for addition/subtraction.
Step 1: 5 x (-2) = -10
Step 2: -3 + (-10) - (-4) = -3 + (-10) + 4
Step 3: -3 + (-10) = -13
Step 4: -13 + 4 = -9
❓Concept Check
Evaluate: |(-3) x 4| - |(-2) + 7|. Show each step.
▸
Concept Check
Evaluate: |(-3) x 4| - |(-2) + 7|. Show each step.
(-3) x 4 = -12, so |(-3) x 4| = |-12| = 12.
(-2) + 7 = 5, so |(-2) + 7| = |5| = 5.
12 - 5 = 7.
Final answer: 7.
Looking Ahead
Integers are whole numbers — they snap to the tick marks on the number line. But real-world quantities rarely land on exact whole numbers. A tank is 3/4 full. A stock drops 2.75 points. A recipe needs 1 1/3 cups of flour. In Unit 2, you will extend everything you learned here to the full set of rational numbers — fractions and decimals included — and the sign rules will carry over unchanged.
