Properties of Numbers

Learn the fundamental rules that govern how numbers behave in arithmetic operations.

Elementary25 minLesson
Printable worksheet

Definition

Properties of numbers are rules that describe how numbers behave when we add, subtract, multiply, or divide them. These properties help us understand patterns and simplify calculations.
The four main properties are:
  1. 1.Commutative Property: The order of numbers doesn't matter
a+b=b+aanda×b=b×aa + b = b + a \quad \text{and} \quad a \times b = b \times a
  1. 2.Associative Property: The grouping of numbers doesn't matter
(a+b)+c=a+(b+c)and(a×b)×c=a×(b×c)(a + b) + c = a + (b + c) \quad \text{and} \quad (a \times b) \times c = a \times (b \times c)
  1. 3.Identity Property: Special numbers that don't change a value
a+0=aanda×1=aa + 0 = a \quad \text{and} \quad a \times 1 = a
  1. 4.Distributive Property: Multiplication distributes over addition
a×(b+c)=a×b+a×ca \times (b + c) = a \times b + a \times c

Try it now

Which property does 7+5=5+77 + 5 = 5 + 7 demonstrate?

Worked Examples

Show that 4+9=9+44 + 9 = 9 + 4
1

Calculate the left side

4+9=134 + 9 = 13
2

Calculate the right side

9+4=139 + 4 = 13
3

Compare both sides

13=1313 = 13 ✓ → Equal!

Common Mistakes

Thinking subtraction is commutative (5−3=3−55 - 3 = 3 - 5)

Why it's wrong: The commutative property only works for addition and multiplication, NOT for subtraction or division.

Correct: 5−3=25 - 3 = 2, but 3−5=−23 - 5 = -2. These are not equal!

Thinking division is associative: (8÷4)÷2=8÷(4÷2)(8 \div 4) \div 2 = 8 \div (4 \div 2)

Why it's wrong: The associative property only works for addition and multiplication, NOT for subtraction or division.

Correct: (8÷4)÷2=2÷2=1(8 \div 4) \div 2 = 2 \div 2 = 1, but 8÷(4÷2)=8÷2=48 \div (4 \div 2) = 8 \div 2 = 4. Not equal!

Confusing identity numbers: Using 1 for addition or 0 for multiplication

Why it's wrong: Adding 1 changes a number, and multiplying by 0 gives 0, not the original number.

Correct: For addition, the identity is 0 (a+0=aa + 0 = a). For multiplication, the identity is 1 (a×1=aa \times 1 = a).

Interactive Visual

Distributive property with numbers

a rows of (b + c) squares split into a rows of b and a rows of c: the total stays the same.

6×(4+3)6 \times (4 + 3)

6×(4+3)=6×4+6×36 \times (4 + 3) = 6 \times 4 + 6 \times 3

=24+18=42= 24 + 18 = 42

Commutative property of addition

Swap the two strips: the total length does not change.

4+94 + 9
9+49 + 4

4+9=9+4=134 + 9 = 9 + 4 = 13

Interactive Sandbox

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Practice Problems

15 problems
Problem 1 of 15
Easy
Which property does 7+5=5+77 + 5 = 5 + 7 demonstrate?

Why It Matters

Understanding number properties helps you:
  • Calculate faster: Rearrange numbers to make mental math easier
  • Check your work: Verify answers using different groupings
  • Solve equations: Apply properties to simplify complex problems
  • Understand algebra: These same properties apply to variables and expressions
For example, instead of calculating 7+38+37 + 38 + 3, you can use the commutative property to reorder it as 7+3+38=10+38=487 + 3 + 38 = 10 + 38 = 48 - much easier!

Real World Applications

Shopping Calculations

Use the commutative and associative properties to add prices quickly.

Example:

Items cost 7 dollars, 5 dollars, and 13 dollars. Instead of adding 7+5+137 + 5 + 13 in order, swap and group: (7+13)+5=20+5=25(7 + 13) + 5 = 20 + 5 = 25 dollars.

1Try It Yourself

You're buying items that cost 8 dollars, 12 dollars, 6 dollars, and 4 dollars.

Rearrange to find the total using friendly numbers.

Step 1: Write the mathematical expression

Group numbers that add to 10 or 20:

Calculating Areas

The distributive property helps calculate areas of combined shapes.

Example:

A garden is 5 meters wide. One section is 8 meters long, another is 2 meters long. Total area: 5×(8+2)=5×10=505 \times (8 + 2) = 5 \times 10 = 50 square meters.

2Try It Yourself

A floor has two sections: 4 meters by 6 meters and 4 meters by 9 meters.

Use the distributive property to find the total area.

Step 1: Write the mathematical expression

Write as: 4×(6+9)4 \times (6 + 9)

Key Takeaways

  • 1Commutative Property: Order doesn't matter for + and ×\times (a+b=b+aa + b = b + a)
  • 2Associative Property: Grouping doesn't matter for + and ×\times ((a+b)+c=a+(b+c)(a+b)+c = a+(b+c))
  • 3Identity Property: a+0=aa + 0 = a and a×1=aa \times 1 = a
  • 4Distributive Property: a×(b+c)=a×b+a×ca \times (b + c) = a \times b + a \times c
  • 5The commutative and associative properties do NOT work for subtraction or division

Frequently Asked Questions

Subtraction and division are not commutative: 5−3≠3−55 - 3 \neq 3 - 5, because the order decides what is taken away from what, and 12÷3≠3÷1212 \div 3 \neq 3 \div 12. They are not associative either: (10−5)−2=3(10 - 5) - 2 = 3, but 10−(5−2)=710 - (5 - 2) = 7. Multiplication does distribute over subtraction, though: a×(b−c)=a×b−a×ca \times (b - c) = a \times b - a \times c.
Subtraction and division are not commutative: 5−3≠3−55 - 3 \neq 3 - 5, because the order decides what is taken away from what, and 12÷3≠3÷1212 \div 3 \neq 3 \div 12. They are not associative either: (10−5)−2=3(10 - 5) - 2 = 3, but 10−(5−2)=710 - (5 - 2) = 7. Multiplication does distribute over subtraction, though: a×(b−c)=a×b−a×ca \times (b - c) = a \times b - a \times c.
Any number multiplied by zero equals zero: a×0=0a \times 0 = 0. This is different from the identity property, which keeps the number unchanged.
The same properties apply to variables! For example, x+y=y+xx + y = y + x (commutative) and 2(x+3)=2x+62(x + 3) = 2x + 6 (distributive). Understanding these makes algebra much easier.

Glossary

Commutative Property
A property stating that changing the order of numbers in addition or multiplication does not change the result
Associative Property
A property stating that changing the grouping of numbers in addition or multiplication does not change the result
Identity Property
Adding 0 or multiplying by 1 leaves a number unchanged
Distributive Property
Multiplying a number by a sum is the same as multiplying by each addend and then adding
Addend
A number being added to another number
Factor
A number being multiplied by another number
Today's challengeOne problem, three tries

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