Properties of Algebra
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Back to Pre-Algebra or back to Algebra I
The properties of algebra are the tools you'll use in order to solve equations and to simplify expressions. Most of what we know of high-school mathematics is governed by these properties, or rules.
The properties include:
- Associative Property
- Commutative Property
- Distributive Property
There are three basic properties of numbers, and you'll probably have just a little section on these properties, maybe at the beginning of the course, and then you'll probably never see them again (until the beginning of the next course). Covering these properties is a holdover from the "New Math" fiasco of the 1960s. While these properties will start to become relevant in matrix algebra and calculus (and become amazingly important in advanced math, a couple years after calculus), they really don't matter a whole lot now.
Why not? Because every math system you've ever worked with has obeyed these properties. You have never dealt with a system where a×b didn't equal b×a, for instance, or where (a×b)×c didn't equal a×(b×c). Which is why the properties probably seem somewhat pointless to you. Don't worry about their "relevance" for now; just make sure you can keep the properties straight so you can pass the next test. The lesson below explains how I kept track of the properties.
More information on these properties of algebra can be found below.
Contents |
The Properties
Associative Property
"Associative" comes from "associate" or "group", so the Associative Property is the rule that refers to grouping. For addition, the rule is:
- a + (b + c) = (a + b) + c
In numbers, this means 2 + (3 + 4) = (2 + 3) + 4. For multiplication, the rule is:
- a(bc) = (ab)c
In numbers, this means 2(3×4) = (2×3)4. Any time they refer to the Associative Property, they want you to regroup things; any time a computation depends on things being regrouped, they want you to say that the computation uses the Associative Property.
In general, the associative property is not applied to subtraction. One must be careful when separating a subtraction or minus sign from the number following it. Typically, a number that is preceeded by a "-" (minus, subtraction, or negative sign) has ownership over that sign. For instance:
which is not equal to
.
You can see that 15 is not equal to 5 because the negative, or subtraction, sign on the 3 was removed in the second part of the above example.
Commutative Property
"Commutative" comes from "commute" or "move around", so the Commutative Property is the one that refers to moving stuff around. For addition, the rule is "a + b = b + a"; in numbers, this means 2 + 3 = 3 + 2. For multiplication, the rule is "ab = ba"; in numbers, this means 2×3 = 3×2. Any time they refer to the Commutative Property, they want you to move stuff around; any time a computation depends on moving stuff around, they want you to say that the computation uses the Commutative Property.
The commutative property only applies to addition and multiplication. However, if you were to consider subtraction to be addition of negatives, then the associative property would apply to subtraction as well. For example:
The same thing applies to division. Division is not generally associative, however if you were to think of division as the multiplication of fractions, then the associative property could be applied. For instance:
Distributive Property
The Distributive Property is easy to remember, if you recall that "multiplication distributes over addition". Formally, they write this property as:
- a(b + c) = ab + ac
In numbers, this means, for example, that 2(3 + 4) = 2×3 + 2×4. Any time they refer in a problem to using the Distributive Property, they want you to take something through the parentheses (or factor something out); any time a computation depends on multiplying through a parentheses (or factoring something out), they want you to say that the computation uses the Distributive Property.
