The commutative property says that changing the order of the numbers does not change the result of adding or multiplying them: a + b = b + a and a × b = b × a. For example, 4 + 9 = 9 + 4 and 3 × 5 = 5 × 3. Subtraction and division are not commutative.
In Spanish
propiedad conmutativa
Also called
commutative law, order property, commutative property of addition, commutative property of multiplication
Turning an array does not change how many dots it has, so 3 × 5 = 5 × 3. Swapping works for + and ×, never for − or ÷.
Formal definition and notation
An operation · is commutative on a set if a · b = b · a for all a and b in the set. For all real numbersa and b, addition and multiplication are commutative: a + b = b + a and a · b = b · a. Subtraction and division are not: in general a − b ≠ b − a and a ÷ b ≠ b ÷ a.
Symbol
Read it as
Meaning
a + b = b + a
a plus b equals b plus a
commutative property of addition
ab = ba
a times b equals b times a
commutative property of multiplication
Key facts
The name comes from “commute”, to move or travel: the numbers swap places.
It halves the facts to memorize: knowing 6 × 8 = 48 gives 8 × 6 = 48 for free.
It works for fractions, decimals, negatives and variables: −2 + 7 = 7 + (−2) and 3x = x · 3.
Rewrite subtraction as adding a negative to reorder safely: 10 − 4 + 6 = 10 + 6 + (−4) = 12.
The commutative property changes order; the associative property changes grouping.
Try it: swap the length and width
Interactive
The rectangle is 6 by 3. Drag the corner to make it 3 by 6: the area stays 18 square units, which is the commutative property of multiplication.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Commutative Property examples, step by step
Example 1: Fill in the blank
Warm-up
Fill in the box: 27 + 58 = 58 + □.
The commutative property swaps the addends: a + b = b + a.
Distributive links multiplication and addition: 3(4 + 5) = 3 × 4 + 3 × 5. Commutative uses one operation at a time.
Quick check: 3 questions
Question 1 of 3
Which equation shows the commutative property?
Show answer and explanation
Answer: C. Only 9 × 4 = 4 × 9 swaps the order of two numbers. 7 × 1 and 8 + 0 are identity properties, and 2(3 + 5) is the distributive property.
Question 2 of 3
Which operation is commutative?
Show answer and explanation
Answer: B. Multiplication is commutative: 2 × 7 = 7 × 2. Subtraction (7 − 2 ≠ 2 − 7) and division (8 ÷ 2 ≠ 2 ÷ 8) are not.
Question 3 of 3
If a + 14 = 14 + 23, what is a?
Show answer and explanation
Answer: D. By the commutative property, 23 + 14 = 14 + 23, so a = 23. 37 is the sum, 14 is already in place, and 9 is 23 − 14.
Commutative Property: frequently asked questions
Is subtraction commutative?
No. 10 − 3 = 7 but 3 − 10 = −7, so swapping the numbers changes the answer. Division is not commutative either: 10 ÷ 2 = 5 but 2 ÷ 10 = 0.2. To reorder a subtraction safely, write it as adding a negative: 10 − 3 = 10 + (−3) = (−3) + 10.
What is the difference between the commutative and associative properties?
The commutative property changes the order of the numbers: 4 × 7 = 7 × 4. The associative property keeps the order but changes the parentheses: (4 × 7) × 2 = 4 × (7 × 2). A memory trick: to commute is to travel, so the numbers move; to associate is to group with friends, so the parentheses change.
Updated November 2026·Reviewed by the Effortless Math team·Standards: CCSS 1.OA.B.3, CCSS 3.OA.B.5, CCSS 6.EE.A.3
Cite this entry
Effortless Math Team. “Commutative Property.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/commutative-property/
Flashcard
TermCommutative Propertya + b = b + a
DefinitionOrder does not change a sum or product: 4 + 9 = 9 + 4, 3 × 5 = 5 × 3. Not true for subtraction or division.
Real NumberA real number is any number that can be placed on the number line. The real numbers are all the rational numbers (fractions, integers, terminating and repeating decimals) together with all the irrational numbers (such as √2 and π). Square roots of negative numbers, like √−4, are not real.Number System · Grade 8, Algebra 1AdditionAddition is the operation that puts amounts together to find the total. The numbers you add are called addends, the answer is the sum, and the sign is the plus sign, +. For example, 7 + 5 = 12. On a number line, adding a positive number means moving to the right.Operations · Kindergarten, Grade 1, Grade 2MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.Operations · Grade 2, Grade 3, Grade 4SubtractionSubtraction is the operation that finds how much is left when you take one amount away from another, or how far apart two numbers are. It is written with a minus sign: 9 − 4 = 5. The answer is called the difference. Subtraction undoes addition, so you can check 9 − 4 = 5 by adding 5 + 4 = 9.Operations · Grade 1, Grade 2, Grade 3DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2Associative PropertyThe associative property says that when you add or multiply three or more numbers, the way you group them does not change the result: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c). It lets you pick the easiest pair first. It does not work for subtraction or division.Operations · Grade 1, Grade 3, Grade 6AddendAn addend is any number that is added to another number. In 8 + 5 = 13, the addends are 8 and 5, and 13 is the sum. An addition can have two or more addends, and you can add them in any order. When one addend is missing, subtract the known addend from the sum to find it.Operations · Grade 1, Grade 2, Grade 3SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2ReciprocalThe reciprocal of a number is 1 divided by that number, so the two multiply to 1. For a fraction, swap the numerator and denominator: the reciprocal of 34 is 43, and the reciprocal of 5 is 15. Zero has no reciprocal. Dividing by a fraction is the same as multiplying by its reciprocal.Number System · Grade 5, Grade 6, Grade 7Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8ArrayAn array is an arrangement of objects in equal rows and equal columns, forming a rectangle. A 4 by 6 array has 4 rows of 6, so it holds 4 × 6 = 24 objects. Arrays show multiplication as equal groups, and turning one sideways shows that 4 × 6 = 6 × 4.Operations · Grade 2, Grade 3, Grade 4Equivalent ExpressionsEquivalent expressions are expressions that have the same value for every value of the variable, even though they look different. For example, 3(x + 2) and 3x + 6 are equivalent: both equal 15 when x = 3 and 0 when x = −2. You build them with the properties of operations, such as the distributive property.Algebra · Grade 6, Grade 7, Algebra 1Combining Like TermsCombining like terms is simplifying an expression by adding or subtracting the terms that have exactly the same variable part. You add their coefficients and keep the variable part: 3x + 2x = 5x. Constants combine with constants. So 3x + 5 + 2x − 1 simplifies to 5x + 4.Algebra · Grade 6, Grade 7, Algebra 1Fact FamilyA fact family is a group of related math facts that use the same three numbers. With 6, 9 and 15 you get 6 + 9 = 15, 9 + 6 = 15, 15 − 6 = 9 and 15 − 9 = 6. Multiplication and division make fact families too, such as 4, 7 and 28. They show that operations undo each other.Operations · Grade 1, Grade 2, Grade 3Identity PropertyThe identity property says a number keeps its value when you add 0 or multiply by 1: a + 0 = a and a × 1 = a. So 0 is the additive identity and 1 is the multiplicative identity. For example, 15 + 0 = 15 and 15 × 1 = 15. It is how equivalent fractions work.Operations · Grade 1, Grade 3, Grade 6Zero Property of MultiplicationThe zero property of multiplication says that any number multiplied by 0 equals 0: a × 0 = 0 and 0 × a = 0. For example, 58 × 0 = 0. If even one factor in a product is 0, the whole product is 0. It explains why 5 ÷ 0 has no answer.Operations · Grade 3, Grade 4, Grade 6