Product (mathematics)
In mathematics, the term "product" refers to the result obtained when two or more numbers, variables, or expressions are multiplied together. Each of the numbers or expressions involved in this operation is known as a factor. The concept of multiplication has a rich historical background, with evidence indicating that ancient civilizations such as the Sumerians and Egyptians utilized multiplication techniques as far back as 3000 BCE and 1650 BCE, respectively. Over the centuries, various mathematicians have contributed to the understanding and formulation of the laws governing multiplication.
Key properties associated with multiplication include the commutative property, which states that the order of factors does not affect the product (e.g., \(a \times b = b \times a\)), and the distributive property, which illustrates that multiplying a sum by a factor can be done by distributing the factor across each addend (e.g., \(a \times (b + c) = a \times b + a \times c\)). Additionally, the associative property indicates that when multiplying three or more factors, the grouping of those factors does not change the product (e.g., \((a \times b) \times c = a \times (b \times c)\)). These foundational concepts are essential for understanding more complex mathematical operations and applications.
On this Page
Subject Terms
Product (mathematics)
When numbers, variables, or expressions are multiplied, the resulting quantity or expression is called the product. Each number, variable, or expression being multiplied is a factor. There are several laws of multiplication that explain the properties of factors and products and how they relate to one another.
Overview
Archeological evidence shows that ancient cultures practiced multiplication to calculate products. The Sumeriansused multiplication and division tables, among other mathematical operations and concepts, around 3000 BCE; the Egyptians used a graphic organizer to help them calculate products using the concept of binary numbers around 1650 BCE; and multiplication was just one of many mathematical operations in use in India before 1000 BCE.
Albertus Magnus (1193–1280 CE), a Dominican bishop and philosopher from Germany, used the term productum in his work Metaphysicorum to refer to the quantity derived from multiplying two or more factors. During the fifteenth century, other mathematicians used the terms product and sum to refer to this quantity, while in the sixteenth century the terms offcome, productus, numerus productus, tota summa, summa producta, and factus were also used to mean the product of multiplication.
In the nineteenth century, mathematicians began to recognize several mathematical laws that explain the properties of products and factors. In 1814, French mathematician François Joseph Servois (ca. 1767–1847) described the commutative and distributive properties of multiplication. The commutative property of multiplication, or the order property of multiplication, states that the product is the same regardless of the order in which the factors are multiplied. An equation representing the commutative property can be written with variables like so: a × b = b × a. According to the distributive property of multiplication over addition, multiplying a sum by a factor yields the same product as multiplying each addend (a number being added) by the factor and adding the products together. Using variables, the distributive property of multiplication over addition might look like this: a × (b + c) = a × b + a × c.
In 1843, Irish mathematician William Rowan Hamilton (1805–65) articulated the associative property of multiplication as part of his work on the algebra of quarternions, or four-dimensional algebra. The associative property of multiplication states that the product of multiplying three or more real numbers is always the same, no matter how the factors are grouped within parentheses. An equation representing the associative property of multiplication can be written using variables for the factors as (a × b) × c = a × (b × c).
Bibliography
Arnold, V. I. Mathematical Methods of Classical Mechanics. 2nd ed. New York: Springer, 2010. Print.
Boyer, Carl B. A History of Mathematics. 3rd ed. Hoboken: Wiley, 2011. Digital file.
Hersh, Reuben. What Is Mathematics, Really? New York: Oxford UP, 2009. Print.
MacNeal, E. Mathsemantics, Making Numbers Talk Sense. New York: Viking, 1994. Print.
Mastin, Luke. The Story of Mathematics. Mastin, 2010. Web. 16 Aug. 2016.
Miller, Jeff. Earliest Known Uses of Some of the Words of Mathematics. Miller, 8 Mar. 2014. Web. 16 Aug. 2016.
Roy, Ranjan. Sources in the Development of Mathematics: Series and Products from the Fifteenth to the Twenty-First Century. Cambridge: Cambridge UP, 2011. Digital file.