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# Realnumbersystems

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### Realnumbersystems

1. 1. The Real NumberThe Real NumberSystemSystem
2. 2. Real NumbersReal NumbersReal numbers consist of all the rational and irrationalReal numbers consist of all the rational and irrationalnumbers.numbers.The real number system has many subsets:The real number system has many subsets:Natural NumbersNatural NumbersWhole NumbersWhole NumbersIntegersIntegers
3. 3. Natural NumbersNatural NumbersNatural numbersNatural numbers are the set of countingare the set of countingnumbers.numbers.{1, 2, 3,…}{1, 2, 3,…}
4. 4. Whole NumbersWhole NumbersWhole numbersWhole numbers are the set of numbers thatare the set of numbers thatinclude 0 plus the set of natural numbers.include 0 plus the set of natural numbers.{0, 1, 2, 3, 4, 5,…}{0, 1, 2, 3, 4, 5,…}
5. 5. IntegersIntegersIntegersIntegers are the set of whole numbers and theirare the set of whole numbers and theiropposites.opposites.{…,-3, -2, -1, 0, 1, 2, 3,…}{…,-3, -2, -1, 0, 1, 2, 3,…}
6. 6. Rational NumbersRational NumbersRational numbersRational numbers are any numbers that can beare any numbers that can beexpressed in the form of , whereexpressed in the form of , where aa andand bb areareintegers, and bintegers, and b ≠ 0≠ 0..They can always be expressed by usingThey can always be expressed by usingterminating decimals or repeating decimals.terminating decimals or repeating decimals.ba
7. 7. Terminating DecimalsTerminating DecimalsTerminating decimals are decimals that containTerminating decimals are decimals that containa finite number of digits.a finite number of digits.Examples:Examples: 36.836.8 0.1250.125 4.54.5
8. 8. Repeating DecimalsRepeating DecimalsRepeating decimals are decimals that contain a infiniteRepeating decimals are decimals that contain a infinitenumber of digits.number of digits.Examples:Examples: 0.333…0.333… 7.689689…7.689689…FYI…The line above the decimals indicate that numberFYI…The line above the decimals indicate that numberrepeats.repeats.9.1
9. 9. Irrational NumbersIrrational NumbersIrrational numbersIrrational numbers are any numbers that cannot be expressedare any numbers that cannot be expressedas .as .They are expressed asThey are expressed as non-terminating, non-repeatingnon-terminating, non-repeatingdecimalsdecimals; decimals that go on forever without repeating a; decimals that go on forever without repeating apattern.pattern.Examples of irrational numbers:Examples of irrational numbers:0.34334333433334…0.34334333433334…45.86745893…45.86745893…(pi)(pi)baπ2
10. 10. Other Vocabulary Associated withOther Vocabulary Associated withthe Real Number Systemthe Real Number System……(ellipsis)—continues without end(ellipsis)—continues without end{ } (set)—a collection of objects or numbers. Sets are{ } (set)—a collection of objects or numbers. Sets arenotated by using braces { }.notated by using braces { }.Finite—having bounds; limitedFinite—having bounds; limitedInfinite—having no boundaries or limitsInfinite—having no boundaries or limitsVenn diagram—a diagram consisting of circles orVenn diagram—a diagram consisting of circles orsquares to show relationships of a set of data.squares to show relationships of a set of data.
11. 11. ExampleExampleClassify all the following numbers as natural, whole, integer,Classify all the following numbers as natural, whole, integer,rational, or irrational. List all that apply.rational, or irrational. List all that apply.a.a. 117117b.b. 00c.c. -12.64039…-12.64039…d.d. -½-½e.e. 6.366.36f.f.g.g. -3-3π
12. 12. To show how these number are classified, use the VennTo show how these number are classified, use the Venndiagram. Place the number where it belongs on the Venndiagram. Place the number where it belongs on the Venndiagram.diagram.π9421−94Rational NumbersIntegersWhole NumbersNaturalNumbersIrrational Numbers-12.64039…11706.36-3
13. 13. SolutionSolutionNow that all the numbers are placed where they belong in theNow that all the numbers are placed where they belong in theVenn diagram, you can classify each number:Venn diagram, you can classify each number:117 is a natural number, a whole number, an integer, and a117 is a natural number, a whole number, an integer, and arational number.rational number.is a rational number.is a rational number.0 is a whole number, an integer, and a rational number.0 is a whole number, an integer, and a rational number.-12.64039… is an irrational-12.64039… is an irrational number.number.-3 is an integer and a rational number.-3 is an integer and a rational number.6.36 is a rational number.6.36 is a rational number.is an irrational number.is an irrational number.is a rational number.is a rational number.π9421−
14. 14. FYI…FYI…When taking the square root of any number that isWhen taking the square root of any number that isnot a perfect square, the resulting decimal will benot a perfect square, the resulting decimal will benon-terminating and non-repeating. Therefore, thosenon-terminating and non-repeating. Therefore, thosenumbers are always irrational.numbers are always irrational.