DrDelMathIntermediate Algebra 4th EditionbyK. Elayn Martin-GayChapter Summary |
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.
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.
Properties of Equality
Properties of Inequality
Law of Trichotomy
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