OnRamps Algebra quiz 1.1.1-1.1.2
Interval Notation
In higher mathematics courses interval notation is often used to describe relevant sets of real numbers on the number line. The reader should become familiar with this notation. These sets can be closed intervals, open intervals, or half-open intervals. They are described below.
For , , and elements of the real numbers:
Intervals
Interval | Meaning | Name |
|---|---|---|
closed interval | ||
open interval | ||
half-open interval | ||
half-open interval |
We will also note that intervals (or any sets for that matter) can be strung together with a union symbol, , which strictly implies a mathematical “or statement.” This means that all the elements are considered in any of the listed sets. For example,
means all elements to be considered belong to the interval or the interval .
The Real Numbers
As previously mentioned, the real numbers consist of all of the rational and irrational numbers. The real numbers are defined as all numbers that have points on the number line. In this sense the real numbers are complete in that they completely fill in the number line. Another useful property of the real numbers is that they constitute a field. The field of real numbers adheres to 11 rules which are also known as axioms. There are five axioms for addition and five for multiplication, as well as one axiom that connects addition and multiplication.
The 11 field axioms for the real numbers: For all , , and :
Field Axioms
Addition Axioms | Multiplication Axioms | |
|---|---|---|
Closure |
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Commutative Rules |
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Associative Rules |
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Identity Elements | Additive Identity Note that the additive identity, 0, is unique in that it is the only real number that has this property under addition. | Multiplicative Identity Note that the multiplicative identity, 1, is unique in that it is the only real number that has this property under multiplication. |
Inverse Rules | , for | |
Distributive Rule | Axiom Connecting Addition and Multiplication | |
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