Closure Property
When you add two whole numbers, do you always get a whole number? Yes! For example, 3 + 5 = 8, and 8 is also a whole number.
This is called the closure property — the set of whole numbers is "closed" under addition because the result always stays in the same set.
But is this true for all operations? In Class 6, you will check closure for addition, subtraction, multiplication, and division.
What is Closure Property - Grade 6 Maths (Whole Numbers)?
Definition: A set of numbers is closed under an operation if performing that operation on any two numbers from the set always gives a result that is also in the set.
If a and b are in the set, and a ⊕ b is also in the set, then the set is closed under ⊕.
Closure Property Formula
Closure of Whole Numbers (0, 1, 2, 3, ...):
Addition — CLOSED:
- Any two whole numbers added give a whole number.
- Examples: 3 + 7 = 10 ✓, 0 + 5 = 5 ✓, 100 + 200 = 300 ✓
Multiplication — CLOSED:
- Any two whole numbers multiplied give a whole number.
- Examples: 4 × 6 = 24 ✓, 0 × 8 = 0 ✓, 15 × 3 = 45 ✓
Subtraction — NOT CLOSED:
- Sometimes subtracting gives a negative number, which is NOT a whole number.
- Example: 3 − 7 = −4 ✗ (−4 is not a whole number)
- But 7 − 3 = 4 ✓. It works sometimes, but not always.
Division — NOT CLOSED:
- Sometimes dividing gives a fraction or is undefined.
- Example: 5 ÷ 2 = 2.5 ✗ (not a whole number)
- Example: 7 ÷ 0 = undefined ✗
Types and Properties
Closure for other number sets:
- Integers (... −2, −1, 0, 1, 2, ...): Closed under addition, subtraction, and multiplication. NOT closed under division (e.g., 1 ÷ 2 = 0.5).
- Natural numbers (1, 2, 3, ...): Closed under addition and multiplication. NOT closed under subtraction (3 − 5 = −2) or division.
Summary table:
- Whole numbers + Addition → Closed ✓
- Whole numbers + Subtraction → Not closed ✗
- Whole numbers + Multiplication → Closed ✓
- Whole numbers + Division → Not closed ✗
Solved Examples
Example 1: Closure under Addition
Problem: Is the sum of 45 and 67 a whole number?
Solution:
45 + 67 = 112. Yes, 112 is a whole number.
Answer: Yes. Whole numbers are closed under addition.
Example 2: Closure under Multiplication
Problem: Is 12 × 15 a whole number?
Solution:
12 × 15 = 180. Yes, 180 is a whole number.
Answer: Yes. Whole numbers are closed under multiplication.
Example 3: Subtraction Fails Closure
Problem: Is 8 − 15 a whole number?
Solution:
8 − 15 = −7. No, −7 is a negative number, not a whole number.
Answer: No. Whole numbers are not closed under subtraction.
Example 4: Division Fails Closure
Problem: Is 7 ÷ 3 a whole number?
Solution:
7 ÷ 3 = 2.33... No, this is not a whole number.
Answer: No. Whole numbers are not closed under division.
Example 5: Closure with Zero
Problem: Is 0 + 0 a whole number? Is 0 × 0 a whole number?
Solution:
- 0 + 0 = 0 ✓ (whole number)
- 0 × 0 = 0 ✓ (whole number)
Answer: Yes, both results are whole numbers.
Example 6: Natural Numbers and Subtraction
Problem: Are natural numbers closed under subtraction?
Solution:
Natural numbers are 1, 2, 3, ...
Consider: 2 − 5 = −3. Not a natural number.
Also: 3 − 3 = 0. Zero is NOT a natural number.
Answer: No, natural numbers are not closed under subtraction.
Example 7: Integers and Subtraction
Problem: Are integers closed under subtraction?
Solution:
Integers include all positive, negative whole numbers and zero.
5 − 8 = −3 (integer ✓). −4 − (−6) = 2 (integer ✓).
Subtracting any two integers always gives an integer.
Answer: Yes, integers are closed under subtraction.
Real-World Applications
Why closure property matters:
- Knowing what to expect: If you add whole numbers, you know the answer will be a whole number — no surprises.
- Algebra: When simplifying expressions, closure tells you the result type stays the same.
- Number system expansion: Because whole numbers are NOT closed under subtraction, we need integers. Because integers are not closed under division, we need fractions.
- Computer programming: Data types in programs follow closure rules — integer + integer = integer.
Key Points to Remember
- Closure property: A set is closed under an operation if the result is always in the same set.
- Whole numbers are closed under addition and multiplication.
- Whole numbers are NOT closed under subtraction (can give negatives) or division (can give fractions).
- To show a set is NOT closed, you only need one counter-example.
- Integers are closed under addition, subtraction, and multiplication (but not division).
- The need for new number systems (integers, fractions) comes from closure failures.
Practice Problems
- Are whole numbers closed under addition? Give two examples.
- Show with an example that whole numbers are NOT closed under subtraction.
- Is 15 ÷ 4 a whole number? What does this tell you about closure?
- Are natural numbers closed under multiplication? Give an example.
- Are integers closed under division? Give a counter-example.
- Fill in: Whole numbers are closed under ______ and ______, but not under ______ and ______.
Frequently Asked Questions
Q1. What does closure property mean?
A set has the closure property for an operation if performing that operation on any two members of the set always gives a result that is also a member of the set.
Q2. Why are whole numbers not closed under subtraction?
Because subtracting a larger number from a smaller one gives a negative result (like 3 − 7 = −4), and negative numbers are not whole numbers.
Q3. Why are whole numbers not closed under division?
Because dividing may give a fraction (like 5 ÷ 2 = 2.5) or be undefined (like 5 ÷ 0). Fractions and undefined results are not whole numbers.
Q4. How do I prove a set is NOT closed?
Find just one example where the operation gives a result outside the set. This one counter-example is enough.
Q5. Are fractions closed under division?
Almost — any non-zero fraction divided by another non-zero fraction gives a fraction. But division by zero is still undefined. So fractions are closed under division except for dividing by 0.
