K-12 – Commutative, Associative, Distributive, Identity and Inverse Laws

Contents

The big four math operations — addition, subtraction, multiplication, and division — let you combine numbers and perform calculations. Certain operations possess properties that enable you to manipulate the numbers in the problem, which comes in handy, especially when you get into higher math like algebra. The important properties you need to know are the commutative property, the associative property, and the distributive property. Understanding an identity and inverse operation is is also helpful.

Law / Property	Addition	Multiplication
Commutative	a + b = b + a	a ⋅ b = b ⋅ a
Associative	(a + b) + c = a + (b + c)	(a ⋅ b) ⋅ c = a ⋅ (b ⋅ c)
Distributive		a(b + c) = ab + ac
Identity	a + 0 = a	a ⋅ 1 = a
Inverse	a + (-a) = 0	a ⋅ (1/a) = 1

Properties of Real Numbers

Commutative Law

Commutative Property of Addition

The commutative property of addition states that when two numbers are being added, their order can be changed without affecting the sum.

For any real numbers a and b, a + b = b + a.

Subtraction is not commutative.

Commutative Property of Multiplication

The commutative property of multiplication states that when two numbers are being multiplied, their order can be changed without affecting the product.

For any real numbers a and b, a ⋅ b = b ⋅ a.

Just as subtraction is not commutative, neither is division commutative.

Associative Law

Associative Property of Addition

The associative property of addition states that numbers in an addition expression can be grouped in different ways without changing the sum. You can remember the meaning of the associative property by remembering that when you associate with family members, friends, and co-workers, you end up forming groups with them.

For any real numbers a, b, and c, (a + b) + c = a + (b + c).

Associative Property of Multiplication

Multiplication has an associative property that works exactly the same as the one for addition. The associative property of multiplication states that numbers in a multiplication expression can be regrouped using parentheses.

For any real numbers a, b, and c, (a ⋅ b) ⋅ c = a ⋅ (b ⋅ c).

Distributive Law

Distributive Property of Multiplication

The product of a sum (or a difference) and a is the same as the sum (or difference) of the product of each addend (or each being subtracted) and the number.

For any real numbers a, b, and c, a(b + c) = ab + ac.

Identity Law

Identity Property of Addition

The identity property of addition is a property of real numbers that states that the sum of 0 and any number is equal to that number.

For any real number a, a + 0 = a.

Identity Property of Multiplication

The identity property of multiplication is a property of real numbers that states that a number retains the same value when multiplied by 1.

For any real number a, a ⋅ 1 = a.

Inverse Law

Inverse Property of Addition

The inverse property of addition is a property of real numbers that states that the sum of a number and its negative (the “additive inverse”) is always zero.

For any real number a, a ⋅ (-a) = 0.

Inverse Property of Multiplication

The inverse property of multiplication is a property of real numbers that states that the multiplication of a number and its reciprocal (the “multiplicative inverse”) is always one.

For any real number a, a ⋅ (1/a) = 1.

Summary

The commutative, associative, and distributive properties help you rewrite a complicated algebraic expression into one that is easier to deal with. When you rewrite an expression by a commutative property, you change the order of the numbers being added or multiplied. When you rewrite an expression using an associative property, you group a different pair of numbers together using parentheses. You can use the commutative and associative properties to regroup and reorder any number in an expression as long as the expression is made up entirely of addends or factors (and not a combination of them). The distributive property can be used to rewrite expressions for a variety of purposes. When you are multiplying a number by a sum, you can add and then multiply. You can also multiply each addend first and then add the products together. The same principle applies if you are multiplying a number by a difference.

References

“9.3.1: Associative, Commutative, and Distributive Properties.” 2021. Mathematics LibreTexts. September 5. https://math.libretexts.org/Bookshelves/Applied_Mathematics/Developmental_Math_(NROC)/09:_Real_Numbers/9.03:_Properties_of_Real_Numbers/9.3.01:_Associative_Commutative_and_Distributive_Properties.

Additional Reading

Remick, Karen. 2022. “A(B+C) = AB + AC.” Medium. Math Simplified. May 23. https://medium.com/math-simplified/a-b-c-ab-ac-585b0a7d3cbd.

When I first was shown A(B+C) = AB + AC (the distributive property) , I was like “OK, that’s obvious. Why are they bothering to show us this? What use is it?” I thought it was yet more of the useless bits of information taught to us to use in mathematical proofs. Little did I know how critical it is to doing math in your head.

Stapel, Elizabeth. 2023. “How to Tell the Basic Number Properties Apart.” Purplemath. Accessed October 27. https://www.purplemath.com/modules/numbprop.htm.

⭐ “What Are the Number Properties? (Commutative, Distributive, Associative & Identity) – BYJUS.” 2022. BYJU’S. July 24. https://byjus.com/us/math/numbers-properties/.

Number properties lay down some rules that we can follow while performing mathematical operations. There are four number properties: commutative property, associative property, distributive property and identity property. Number properties are only associated with algebraic operations that are addition, subtraction, multiplication and division. However, some of these properties are not applicable to subtraction and division operations.

⭐ I suggest that you read the entire reference. Other references can be read in their entirety but I leave that up to you.

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