Why is it true that all irrational numbers are non-terminating/non-repeating decimals?

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jbaekk7q · Accepted Answer

The definition: a number is irrational if and only if it&#039;s not rational, i.e. it can&#039;t be expressed as a ratio of two integers. If   x has a repeating decimal expansion (this includes terminating decimal expansions), then   x is rational.Proof: If   x has a repeating decimal expansion, then it can always be written in the following form:Let   c  ,  b be non-negative integers and       a    i    ∈  {  0  ,  1  ,  2  ,  …  ,  9  } and   t is the number of digits of   b.  x  =            c      .      b              a        1                    a        2            …              a        k                    a        1                    a        2            …              a        k                    a        1                    a        2            …        ¯        10    t    x  =            c      b      .              a        1                    a        2            …              a        k                    a        2                    a        2            …              a        k                    a        1                    a        2            …        ¯        10          k      t        x  =            c      b              a        1                    a        2            …              a        k            .              a        1                    a        2            …              a        k                    a        1                    a        2            …        ¯        10          k      t        x  −      10          t        x  =            c      b              a        1                    a        2            …              a        k              ¯    −            c      b        ¯    x  =                              c          b                      a            1                                a            2                    …                      a            k                          ¯            −                        c          b                ¯                            10                  k          t                    −              10        t

Why is it true that all irrational numbers are non-terminating/non-repeating decimals?

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