5. Consider the number x=13.58585858585858... (repeating) a) Explain completely,

Question

5. Consider the number x=13.58585858585858... (repeating)

a) Explain completely, using words, math symbols, and/or diagrams, what this notation really means. How does one make snese out of this infinite decimal, what expansion? How do we get a handle on what number we are talking about here? I'm not asking what the bar means - this is a question about the meaning of the mathematical idea, not about the notation. [HINT: it is not equal to, and does not get close to, a terminating decimal.]

b) If this number x is rational, express it as a/b for natural numbers a and b (obviously, you need to show how, not just use a calculator button). If it is irrational, explain clearly what it cannot be expressed in this way.

Please Show Work, Thank You!

Explanation / Answer

(a) X= 13.5858585858..............................

= 13+ 58/100 + 58/10000+ 58/1000000+..........................

= 13+ 58( 1/100+1/10000+1/1000000+.............................

= 13 + 58 (1/102 + 1/104+ 1/106+.................................( IT IS A GEOMETRIC PROGRESSION WITH COMMON RATIO 1/102 AND FIRST TERM 1/102)

= 13 + 58 [(1/102)/ (1- 1/102 ) = 13 + 58 [ (1/100)/ (99/100)] = 13 + 58 [1/100 X100/99]

= 13 + 58 X 1/99 = ( 1287 + 58)/99 = 1345/99 , NUMBER 1345 DIVIDED BY 99 IS 13.58585858.........

(b) AS THIS NUMBER CAN BE EXPRESSED IN P/Q FORM WHERE` P` AND `Q` ARE NATURAL NUMBER

HENCE IT IS RATIONAL NOT IRRATIONAL.

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