Section 1.1: The Real Number System
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Sets
The Real Number System
Section 1.2: Operations with Real Numbers
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Get in the habit of refering to the expression located between the two vertical bars as the domain element.
This habit will help you understand absolute value a little better.
This habit will help you a great deal when you are introduced to the concept of function.
If we translate the above symbolic definition of absolute value into a verbal statement, we obtain the following three statements:
The Binary Operations in the Real Number system
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.
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 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".
Division of fractions is exactly like all divisions in mathematics
Because the multiplicative inverse of a fraction is its reciprocal, the conversion to multiplication looks like:
After converting to multiplication and performing the multiplication according to the definition of multiplication we get the following rule.
All of this boils down to a fairly simple rule for working with division of fractions as set forth in the following definition.
Subtraction of fractions is exactly like all subtractions in mathematics
Using one of the four forms for the opposite of a fraction makes the conversion to addition look like:
From which we get the following definition of subtraction for fractions.
The product of two real numbers with the same signs is positive.
The product of two real numbers with different signs is negative.
Section 1.3: Powers, Square Roots, and Order of Operations
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Definition: A number b is a square root of a if b2 = a
FACT: Every positive number has two square roots, one that is positive and one that is negative.
Definition: The principal square root of a number is its positive square root. The principale square root of 0 is 0.
FACT: The square root of a negative number is not a real number.
The Pythagorean Theorem: If the length of the hypotenuse of a right triangle is c and the lengths of its legs are a and b, then a2 + b2 = c2
Definition: A number b is a cube root of a if b3 = a
Definition: A number b is the nth root of a if bn = a.
Section 1.4: Integer Exponents and Scientific Notation
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Section 1.5: Operations with Variables and Grouping Symbols
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Section 1.6: Evaluating Variable Expressions and Formulas
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