Realnumbersystemnotes
- 1. The Real Number System Mr. Ferguson’s Class
- 2. Real Numbers Real numbers consist of all the rational and irrational numbers. The real number system has many subsets: Natural Numbers Whole Numbers Integers
- 3. Natural Numbers Natural numbers are the set of counting numbers. {1, 2, 3,…}
- 4. Whole Numbers Whole numbers are the set of numbers that include 0 plus the set of natural numbers. {0, 1, 2, 3, 4, 5,…}
- 5. Integers Integers are the set of whole numbers and their opposites. {…,-3, -2, -1, 0, 1, 2, 3,…}
- 6. Rational Numbers Rational numbers are any numbers that can be expressed in the form of , where a and b are integers, and b ≠ 0 . They can always be expressed by using terminating decimals or repeating decimals.
- 7. Terminating Decimals Terminating decimals are decimals that contain a finite number of digits. Examples: 36.8 0.125 4.5
- 8. Repeating Decimals Repeating decimals are decimals that contain a infinite number of digits. Examples: 0.333… 7.689689… FYI…The line above the decimals indicate that number repeats.
- 9. Irrational Numbers Irrational numbers are any numbers that cannot be expressed as . They are expressed as non-terminating, non-repeating decimals ; decimals that go on forever without repeating a pattern. Examples of irrational numbers: 0.34334333433334… 45.86745893… (pi)
- 10. Other Vocabulary Associated with the Real Number System … (ellipsis)—continues without end { } (set)—a collection of objects or numbers. Sets are notated by using braces { }. Finite—having bounds; limited Infinite—having no boundaries or limits Venn diagram—a diagram consisting of circles or squares to show relationships of a set of data.
- 11. Venn Diagram of the Real Number System Irrational Numbers Rational Numbers
- 12. Example Classify all the following numbers as natural , whole , integer , rational , or irrational . List all that apply. 117 0 -12.64039… -½ 6.36 -3
- 13. To show how these number are classified, use the Venn diagram. Place the number where it belongs on the Venn diagram. Rational Numbers Integers Whole Numbers Natural Numbers Irrational Numbers -12.64039… 117 0 6.36 -3
- 14. Solution Now that all the numbers are placed where they belong in the Venn diagram, you can classify each number: 117 is a natural number, a whole number, an integer, and a rational number. is a rational number. 0 is a whole number, an integer, and a rational number. -12.64039… is an irrational number. -3 is an integer and a rational number. 6.36 is a rational number. is an irrational number. is a rational number.
- 15. FYI…For Your Information When taking the square root of any number that is not a perfect square, the resulting decimal will be non-terminating and non-repeating. Therefore, those numbers are always irrational.