By Bert Markgraf
July 22, 2023 9:42 pm EST
A product is the result of multiplying two or more numbers. Key multiplication properties that simplify calculations are the commutative, distributive, associative, and identity properties. These rules apply to all real numbers, from integers to fractions.
The product of numbers is the value you obtain after performing multiplication. For example, the product of 2, 5, and 7 is:
2 × 5 × 7 = 70
Although different sets of numbers can produce the same product—6 × 4 = 24, 2 × 12 = 24, 8 × 3 = 24—the operation of multiplication is governed by four distinct properties that set it apart from addition, subtraction, and division.
Commutativity means the order of the factors does not affect the product. Whether you compute 8 × 2 or 2 × 8, the result is always 16. This property holds for addition as well but not for subtraction or division.
Examples:
3 ÷ 4 = 0.75 ≠ 4 ÷ 3 = 1.33…
7 – 5 = 2 ≠ 5 – 7 = –2
Multiplying a sum by a number is equivalent to multiplying each addend individually and then adding the results:
4 × (3 + 6) = (4 × 3) + (4 × 6) = 12 + 24 = 36
Division does not share this property: 6 ÷ (3 + 9) ≠ 6 ÷ 3 + 6 ÷ 9.
When multiplying more than two numbers, you can group them arbitrarily without changing the outcome. For example:
12 × (4 × 2) = 12 × 8 = 96
or
(12 × 4) × 2 = 48 × 2 = 96
In contrast, division and subtraction are not associative.
Multiplying any number by the identity element 1 leaves it unchanged:
a × 1 = a
Example:
((24 × 3) + 2 – 6) × 1 = ((24 × 3) – 4) = 68
Understanding the roles of each number helps avoid confusion:
Beyond elementary arithmetic, products appear in various mathematical contexts:
Each of these relies on the core idea of combining factors—what we call multiplicands and multipliers in elementary terms.
For a deeper dive into each type, consult specialized texts in set theory, linear algebra, or tensor analysis.
