Arithmetic Operations

Mathematics education typically begins with a general introduction to the concept of numbers and how those numbers relate to each other, but quickly moves beyond the mere existence of numbers to the idea that these numbers can act upon each other in a variety of ways.

Counting appears to be nearly as old as the ancient civilizations of China, India, and Mesopotamia. These civilizations developed written systems for representing numerical values and subsequently they all also developed methods for implementing the four basic arithmetic operations: addition, subtraction, multiplication, and division.

The mathematical sophistication of these systems evolved over the years as the original civilizations fell and new civilizations arose. While the notation and algorithms changed with the numbering system, the basic arithmetic operations held constant. Rules about their behavior came to be understood and incorporated into the notation system. These arithmetic properties are a combination of observed behaviors of numbers and mathematical conventions based around the notation.

The associative property and distributive property were included in the Elements by Euclid (c. 300 BCE). Medieval universities focused their classical education on the quadrivium, a curriculum built around arithmetic, geometry, music, and astronomy.

Associative Property

The associative property of addition indicates that when computing a series of addition operations in a row with each other, the order in which the operations are performed will not change the outcome. The resulting sum of the expression 2 + 3 + 4 will be identical whether one begins on the left by computing 2 + 3 or on the right by computing 3 + 4.

Multiplication has a well-known interpretation as repeated addition, and so it is not surprising to find that an associative property of multiplication applies when multiplying a series of factors together to obtain a product. Either order in which one performs the steps of the multiplication yields the same solution.

Extending this notion into the inverse operations of subtraction and division can be confusing, because the associative property cannot be applied in as straightforward a way as with either addition or multiplication. If one uses negative numbers, they may translate an expression containing addition and subtraction symbols into one with addition symbols only (5 – 3 + 6 = 5 + –3 + 6)), which then allows the associative property to apply to the positive and negative numbers within this expression. A similar process can be applied with division and fractions.

Commutative Property

The commutative property applies in cases where there are only two terms, indicating that the order of the terms do not change the outcome. As with the associative property, the commutative property applies to both addition and multiplication, and can be expressed in general terms as:

If one did not know the associative property and merely worked from left to right in order of operations, the following example shows that operating on a and b first is equivalent to operating on b and c first, precisely as the associative property indicates. In this way, the commutative property allows for a direct proof of the associative property.

Again, the inverse operations of subtraction and division do not possess this property in a straightforward way. It is not true that 5 – 3 = 3 – 5, though these inverse operations can be transformed into addition or multiplication statements so that the commutative property may be applied, as in 5 + 3 = 3 + 5.

Order of Operations

The above properties apply to expressions that contain only addition or multiplication, with no mix of the two operations within the same expression. Mathematical notation allows for more sophisticated expressions than this, and interpreting this notation properly requires an understanding of the order of operations that must be applied.

A common mnemonic device to remember the order of operations is "Please Excuse My Dear Aunt Sally," abbreviated as PEMDAS, in which the first letter of each word serves as a reminder of the order in which the operations should be executed:

Parentheses

Exponents

Multiplication

Division

Addition

Subtraction

The mnemonic is helpful in remembering the order of operations, but is slightly misleading, as the mnemonic seems to indicate that all multiplication takes place before one begins division, and then all addition before any subtraction. Someone applying the mnemonic with this understandable but incorrect interpretation would arrive at the conclusion that 5 – 3 + 2 = 5 – 5 = 0, rather than the correct order of operations that yields 5 – 3 + 2 = 2 + 2 =4. The multiplication/division steps and the addition/subtraction steps must be treated not as four discrete steps, but rather as two steps that each contain two operations. The multiplication and division step is jointly performed from left to right and then the addition and subtraction step is jointly performed from left to right.

As indicated in the mnemonic, operations within parentheses take priority over the operations outside the parentheses. For example, the expression 5 − 8 ÷ (1 + 3) indicates that the 1 + 3 operation should be performed before the division operation is executed.

5 – 8 ÷ (1 + 3) = 5 – 8 ÷ 4 = 5 – 2 = 3

The inclusion of parentheses helps explicitly write some of the previous concepts described, as it provides a means of noting which operations are conducted before which other operations. The associative property for addition can therefore be expressed in the general form (a + b) + c = a + (b + c).

Distributive Property

The distributive property is in a sense a property that gives permission to circumvent the established order of operations. The order of operations normally require one to evaluate expressions within the parentheses before evaluating any operations outside of the parentheses. The distributive property, however, establishes an equivalency that can be applied to multiply a factor across all terms within the parenthetical before the addition is performed.

a(b + c) =ab + ac

Since c can be negative, by extension this means that the distributive property also applies if the operation is subtraction instead of addition. Division, rather than multiplication, can be represented with a having fractional values.

The equivalence relationships from the earlier properties also apply, as in the case of using the commutative property to establish a(b + c) = ab + ac = ba + ca = (b + c)a. This shows that the common factor of a can be factored out from ab + ac, which provides the basis for factoring, an essential concept throughout algebra and later math applications.

Bibliography

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Kinzer, Cathy J., and Ted Stanford. "The Distributive Property: The Core of Multiplication." Teaching Children Mathematics 20.5 (2013): 302-309. Print.

McKellar, Danica. Kiss My Math: Showing Pre-algebra Who’s Boss. New York: Hudson Street, 2008. Print.

Silver, Zoe. "Order of Operations Bingo." Illuminations: Resources for Teaching Math. National Council of Teachers of Mathematics. Web. 2 Jan. 2015 <http://illuminations.nctm.org/Lesson.aspx?id=2583>.

Strogatz, Steven H. The Joy of X: A Guided Tour of Math, from One to Infinity. Boston: Houghton, 2012. Print.

Tent, Margaret. "Understanding the Properties of Arithmetic: A Prerequisite for Success in Algebra." Mathematics Teaching in the Middle School 12.1 (2006): 22-25. Print.