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Answer:

  DecimalBinary so far
Start 0.750 0.
×21.50 0.1
erase .50  
×21.00 0.11
erase .00  
Result   0.11

So 0.7510 = 0.112.

To check this, go in the other direction: 0.112 = 2-1 + 2-2 = 0.5 + 0.25 = 0.75


Non-terminating Result

  DecimalBinary So Far
Start 0.1 0.
×20.2 0.0
×20.4 0.00
×20.8 0.000
×21.6 0.0001
  .6 0.0001
×2 1.2 0.00011
  0.2 0.00011
×20.4 0.000110
×20.8 0.0001100
×21.6 0.00011001
  .6 0.00011001
×21.2 0.000110011
  0.2 0.000110011
×20.4 0.0001100110
×20.8 0.00011001100
Result   0.00011001100...










At right the algorithm is used to convert 0.110 to binary. (The "erase" steps are not shown when a 0 is copied to the "binary so far" column.)

The algorithm does not end. After it has started up, the same pattern 0.2, 0.4, 0.8, 1.6, 0.6, 1.2, 0.2 repeats endlessly. The pattern 0011 is appended to the growing binary fraction for each repetition.

Unexpected Fact: The value "one tenth" cannot be represented precisely using a binary fraction.

This is true in the base two positional notation used here, and also in floating point representation used in programming languages. This is sometimes an important consideration when high accuracy is needed.


QUESTION 15:

Can "one third" be represented accurately in decimal?