Rational Numbers and Irrational Numbers - M. Olsen's Website



Goals: To discuss division with decimalsRational Numbers and Irrational numbersTerminating and repeating decimals vs non repeating not terminating decimalsConverting fractions to decimalsConverting decimals to fractionsTerminating decimals to fractionsRepeating decimals to fractionsProblems involving fractions and decimals together.Rational Numbers and Irrational NumbersThe set of numbers we call the Real Numbers, R, consist of the union of the sets of both the Rational Numbers, Q fractions, and their compliment the Irrationals. Using a Venn Diagram, it looks like this:left381000These two sets the Rational Numbers and the Irrational numbers have VERY distinct decimal representations that are their “Calling Cards” for their membership in each set. Click here to see a video of explaining this.They are as follows:491998067945Every Irrational Number will have a decimal representation that will simply: Never terminate & Never Repeat!!!Examples: π=3.14159265359…e=2.71828182845… 2=1.4142135… ?=1.61803… 00Every Irrational Number will have a decimal representation that will simply: Never terminate & Never Repeat!!!Examples: π=3.14159265359…e=2.71828182845… 2=1.4142135… ?=1.61803… 51758571575Every Rational Number will have a decimal representation in one of two forms:00Every Rational Number will have a decimal representation in one of two forms:-1045951782551. A terminating decimal(in reduced form, the denominator always has a prime factorization of only 2’s and/or 5’s)Examples: 12=0.573100=0.73 6025=2.41315=26.2 001. A terminating decimal(in reduced form, the denominator always has a prime factorization of only 2’s and/or 5’s)Examples: 12=0.573100=0.73 6025=2.41315=26.2 17759631782542. A non-terminating decimal, BUT it repeats after some point.(denominator has prime factors other than just 2’s and 5’s)Examples: 12=0.573100=0.73 6025=2.41315=26.2 57=714285 002. A non-terminating decimal, BUT it repeats after some point.(denominator has prime factors other than just 2’s and 5’s)Examples: 12=0.573100=0.73 6025=2.41315=26.2 57=714285 This is the real numbers all mixed together on the real number line.center21134700Converting Fractions to DecimalsTo convert a fraction into a decimal you only have to recognize that fractions are simply little division problems! So…divide!!Examples:89 17391556810 84100911000555100Converting decimals to fractionsNot all decimals are fractions, like 3.14159265359….So only the terminating and the repeating (non-terminating) decimals are able to be expressed as fractions.Converting terminating decimals into fractions If you say the proper name of the decimal, you can write it in its fraction or mixed fraction form and thereby express it as a mixed fraction.Examples: Notice, that these decimals will always have a denominator consisting of only 2’s and/or 5’s.0.50.60.1230.95475.56963.851Converting non-terminating but repeating decimals into fractions (skip if you dare)This is not your buddies section, this is for those of us that care enough to learn something new.This is not always easy, it requires us to remember that denominators of numbers consisting of primes other than 2’s and 5’s will always make a decimal repeat, and that 9’s in the denominator are one way to make a number repeat. Consider: Notice the patterns with denominators containing 9’s. Check your fractions with a calculator to be sure they are equal to these decimals.0.10.20.30.40.50.60.70.80.130.760.530.640.850.1430.7120.2530.3640.485But what about decimals like these:Recall: 0.2310=0.023 and 1910=190=0.010.010.020.030.0530.064Challenge problems:0.730.6531.732.653Problems Involving Fractions and Decimals TogetherFractions and terminating or repeating decimals are different formats for expressing the same thing.If you have a problem that has two rational numbers but in different forms, you can pick which form you wish to work with, fraction form or decimal form. Some ways may seem easier than other ways so choose based on the problem.Examples:12+0.250.6-1434+3.7 -310+8.1 2.3-56116-0.3750.6+19 0.4+290.7-59 ................
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