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Algebra - Rational and Irrational Numbers










































Rational and Irrational Numbers

All Real Numbers can be plotted on a number line. Real Numbers include the set of all rational numbers and the set of all irrational numbers.

Which of the following numbers listed are not rational?

7.2, 5/9, 100², 101²

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Aug 25, 2012
Rational and Irrational Numbers
by: Staff

Answer:
Part I

Although both rational numbers and irrational numbers are both classified as real numbers, they have different characteristics.

• A RATIONAL NUMBER is a number which can be written as a ratio.

       The first 5 letters of the word rational spell the word RATIO, which stands for fraction. A rational number can be written as the quotient of two integers (the denominator cannot be equal to zero).

       All four of the numbers (7.2, 5/9, 100², and 101²) which are given in the problem statement are rational numbers. Each number can be written as the quotient of two integers (a fraction):

          7.2 = 36/5

          5/9

          100² = 10000/1

          101² = 10201/1


       The decimal equivalent of every rational number (fraction) has one of the following two patterns:

              A terminating decimal (a decimal that does not go on forever).

              A repeating decimal pattern (a decimal that repeats the same pattern over and over, forever).

       The decimal equivalent of all four of the numbers (7.2, 5/9, 100², and 101²) which are given in the problem statement are:


          7.2 (terminating decimal)

          5/9 = 0.555555555555… (repeating decimal)

          100² = 10000. (terminating decimal)

          101² = 10201. (terminating decimal)

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Aug 25, 2012
Rational and Irrational Numbers
by: Staff

---------------------------------------------

Part II

• An IRRATIONAL NUMBER is a number which cannot be written as a fraction of two integers a/b.


       In addition, the decimal value of an irrational number is either undefined, or a non-terminating decimal which does not have a pattern which repeats itself.

          An example of an irrational number is the square root of 5/0.

             5/0 = undefined

          Another example of an irrational number is the √11.

             √11 ≈ 3.316624790355399849114932736670686683927088545589353597058682146116484642609 etc.


          A third example of an irrational number is the value of pi.

             pi (π) ≈ 3.14159265358979323846264338327950288419716939937510 etc.

       The second and third examples are numbers which do not have a decimal pattern which repeats itself, no matter how many decimal places are computed.

       One way to remember how to recognize an irrational number is to remember that an irrational number behaves like an irrational person: there is NO PATTERN to their behavior – what they say or do unpredictable.


So where did irrational numbers originate?

Certain real numbers cannot be calculated using the normal rules of arithmetic in our base 10 number system. These numbers can only be approximated. The square root of 2 is an example.

This same problem appears in other number base systems as well. For example, 0.1 is a rational number in our base 10 number system. 0.1 can be written as the ratio 1/10 in the base 10 number system.

However, 0.1 is an irrational number in the binary (base 2) number system. It cannot be written as a ratio in the binary base system.

History leaves us this interesting story regarding the Greek mathematician Pythagoras regarding the discovery of irrational numbers (actually, mathematicians in India already knew about irrational numbers – but the story about Pythagoras is a better story).

When one of Pythagoras' students demonstrated that √(2) cannot be expressed as a quotient of two integers, Pythagoras pronounced that this was “irrational”, and commanded the student to be drowned. (http://news.softpedia.com/news/Where-Do-Numbers-Come-From-22456.shtml)





Thanks for writing.

Staff
www.solving-math-problems.com



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