HOW CAN 0.999... = 1 ?????

                                     



We usually round off numbers, and we do say 0.99999=1. How can 1 equal 0.999...? Well, it does, and we can prove that in two different ways.

Proof 01:

      let  N=0.999,

   thn10N=9.999

⟹     (10N - N)  = (9.999 - 0.999)

⟹              9N   = 9.000

⟹                N   = 1


Proof 02:

     The number "0.9999..." can be "expanded" as:

    0.9999... = 0.9 + 0.09 + 0.009 + 0.0009 + ...

In other words, each term in this endless summation will have a "9" preceded by some number of zeroes. This may also be written as:

0.999... = 9/10 + 9/100 + 9/1000 .......

             = 9/10 + 9/10(1/10) + 9/10(1/100) .......

             = 9/10 + 9/10(1/10)¹ + 9/10(1/10)² + 9/10(1/10)³ ....

    That is, this is an infinite geometric series with first term

     a =  9/10 and common ratio r =  1/10. Since the size of the common ratio r is less than 1, we can use the infinite-sum formula to find the value:

      

    0.999 = (9/10)[1/(1-1/10)]

              = (9/10)[1/(9/10)]

              = (9/10) (10/9)

              = 1.

    So the formula proves that 0.9999... = 1.

      


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