Introduction to Commutative Algebra
Algebra is simply logic. Algebraic reasoning is based on consistencies and rules that apply when dealing with numbers or numeric values. Learning and comprehending the properties of numbers and how they behave and interact make memorization less important in the study of algebra. When the basic laws of algebra logic are understood and applied, then what may look like complicated mathematical equations become much easier to solve. One of rules of algebra is the commutative property.
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Definition
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Merriam-Webster's online dictionary defines "commutative" as an adjective that describes something that shows commutation.
"Commutation" is defined as a noun. A commutation is an exchange, substitution or switch.
The commutative property in algebra and mathematics refers to situations in which the order of numbers (or letters) can be switched without affecting the solution.
Commutative Property of Addition
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The following equations all have the same solution:
3 + 4 + 12 + 1 + 4 = 24
12 + 4 + 4 + 1 + 3 = 24
4 + 1 + 3 + 4 + 12 = 24.
The only thing that changes from equation to equation is the order in which the numbers are placed. The numbers being added keep switching positions. Even though the numbers switch positions, the end result (24) is the same.
Addition is commutative because you can switch the order of numbers and always get the same answer.
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Commutative Property of Multiplication
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Multiplication is commutative. For example,
3 x 5 = 15
5 x 3 = 15
3 x 5 = 5 x 3.
You can switch the order of numbers in multiplication regardless of how many numbers are being multiplied, and the end result will still be the same answer.
2 x 6 x 10 x 2 = 240
2 x 2 x 6 x 10 = 240
10 x 2 x 6 x 2 = 240.
Division, however, is not commutative.
10 / 5 = 2, but 5 / 10 = 1/2
9 / 3 = 3, but 3 / 9 = 1/3.
There is a pattern, but there is no commutative property because the answers are different.
Subtraction and Negative Numbers
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Subtraction is not commutative. For example,
6 - 4 = 2
4 - 6 = (-2).
Positive 2 and negative 2 are different values.
We can, however, convert the two subtraction equations into addition equations:
6 - 4 = 2 is also 6 + (-4) = 2 and (-4) + 6 = 2
4 - 6 = (-2) is also 4 + (-6) = (-2) and (-6) + 4 = (-2).
Note the important difference. In the subtraction equations, there is no commutative property.
When we add negative values, we are performing addition. Addition is commutative whether the values are positive or negative.
Expert Insight
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Because the rules of algebra apply to any numeric value, these values can be substituted with letters to demonstrate the logic. For example, the commutative property of addition is often stated as follows:
a + b = b + a.
The commutative property of multiplication is often stated as follows:
xy = yx.
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