DrDelMath

Chapter 1: Real Numbers and Algebraic Expressions

Section 1.1: Tips for Success in Mathematics

In addition to the suggestions in the textbook, you should also read the following essay on this website.

Studying Mathematics

Section 1.2: Algebraic Expressions and Sets of Numbers

You should begin your study of Algebra with a careful study of the sets of Real numbers. To do that click HERE.

Section 1.3: Operations on Real Numbers

With advanced mathematics, it can be proven that the points on an infinitely long line are in one-to-one correspondence with the real numbers. Remember the real numbers include rationals, irrationals, integers, whole numbers, and natural numbers. To say there is a one-to-one correspondence between the points of the line and the real numbers means that:
(i) for every real number there corresponds exactly one point on the line
(ii) for every point on the line there corresponds exactly one real number.

Addition of Real Numbers

This give rise to the concept of "The Number Line" which is a geometric way of visualizing the real numbers.
It is common to draw the number line showing only the integer points. When that is the case it is your responsibility to recognize that all the rational numbers and all the irrational numbers are represented on that same line.

It is imperative that you observe the order of the real numbers as displayed by the number line.
At any point on the number line, the numbers increase to the right and decrease to the left.

This leads to a natural interpretation and visualization for addition of signed numbers. Now note that adding a positive number (to any number) means to increase and thus means move right. On the other hand, adding a negative number (to any number) means to decrease and thus means move left. This visualization will always yield the correct sign for the sum.

On the number line the symbol < (read as "less than") means "to the left of".
On the number line the symbol > (read as "greater than") means "to the right of".

Subtraction of Real Numbers

Multiplication of Real Numbers

Division of Real Numbers

Order of Operations

Section 1.4: Properties of Real Numbers

Some very important properties of the Real numbers are presented below. Overt use of these properties will vary. Sometimes it will be necessary to explicitly reference one of these properties and on other occasions, the property will be used with hardly a notice.

The following properties of addition and multiplication are presented as properties of Real Numbers. Consequently they are properties of Natural Numbers, Whole Numbers, Integers, Rational Numbers, Irrational Numbers, and Real Numbers.

Axioms for the Real Number System

Properties of Equality

Properties of Inequality

Law of Trichotomy