Most textbooks for early algebra courses (Algebra I, Algebra II, Beginning, Intermediate, and College Algebra) present quite a few axioms and other elementary properties of the Integers and later the Real Numbers. These presentations are usually accompanied with brief discussions and examples. Unfortunately, most textbooks do not explicitly make use of these properties beyond the initial introduction. And they do not teach students when and how to use the properties.
Following is a partial list of these properties. These properties are usually presented in symbolic form. Because my audience here is not restricted to mathematics students or teachers, I will present the properties in plain English. All teachers, at all levels, of mathematics should know these properties and the related Algebraic structures.
The Number Line
Discussion of real numbers, at every possible opportunity, should be illustrated on the number line.
Caveat: The number line should be introduced only at an age-appropriate time in student development.
Complex numbers should be illustrated in the plane (Cartesian Coordinate system).
Closure: The sum or product of two integers is an integer.
Uniqueness: The sum or product of equals are equal.
Commutative Laws: The sum or product of two integers is independent of their order.
Associative Laws: The sum or product of three integers is independent of their grouping.
Distributive Law: The product of an integer and a sum of integers is equal to the sum of the product of that integer and each of the summands.
Zero: There is a unique integer whose sum with any other integer is that same integer.
Unity: There is a unique integer whose product with any other integer is that same integer.
Additive Inverse: Each integer has an opposite with the property that the sum of an integer and its opposite is zero.
Zero Factor Property: If the product of two integers is zero, one of the factors is zero.
Rules of Equality
Reflexive Law: Each integer is equal to itself.
Symmetric Law: Equality of two integers does not depend on order of presentation.
Transitive Law: If two integers are each equal to a third integer, the two integers are equal.
Properties of Order
Addition: The sum of two positive integers is positive.
Multiplication: The product of two positive integers is positive.
Law of Trichotomy: An integer satisfies exactly one of the following:
- It is greater than zero
- It is zero
- It is less than zero
Property of Division
The Euclidean Algorithm: If an integer (dividend) is divided by as non-zero integer (divisor) there is a unique integer quotient and integer remainder such that the dividend is equal to the sum of the remainder and the product of the divisor and quotient. Furthermore, the remainder is non-negative and is less than the divisor.
Properties of Rational, Real, and Complex Numbers
Previous Properties: Each of the above axioms and properties, except for the Euclidean Algorithm are true for Rational, Real, and Complex Numbers.
Multiplicative Inverse: Each non-zero number has a reciprocal with the property that the product of a number and its reciprocal is the unity.
All teachers, at all levels, of mathematics should know these properties and the related Algebraic structures. I am not advocating that all mathematics teachers should present these properties to their students. In fact, I would not present this list until quite late (after high school). However, teachers should possess this knowledge to enable them to teach with the proper perspective. For example, teachers should emphasize that zero is a very special number with very special properties – there is only one number like it. As another example, teachers should help students to understand the similarities between 0 and 1 – the difference is the operation. Another example is to use the number line to drive home the fact that there are irrational numbers.
I believe we may assume students correctly use most of these properties at appropriate times in early college algebra courses. However, during the last decade or more of my teaching career I identified four properties that require explicit and deliberate attention. I identified four properties that are poorly understood and have frequent application in these early college algebra courses. As a means of attracting attention and separating my discussion from the past I grouped these four properties under the group name “The Big Four.”
The Big Four consists of The Distributive Property. The Transitive Property, The Law of Trichotomy, and The Zero Factor Property.
I plan to discuss The Big Four in considerable detail in the next series of posts.