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Dedekind's Kingdom for a Definition of "Exact"


Consider these "exact" sums:

  • 2 + 3
  • 1/2 + 1/5
  • 1/3 + 1/5
  • 1/3 + 1/7
Task 1: Is Pi + sqrt(2) an "exact" sum?

Task 2: Is 0.404004000400004... + 0.17727773777747777757... an "exact" sum?

Task 3: In the sum 0.404004000400004... + 0.17727773777747777757... , what is the 1000th digit? 1,000,000th digit?

 

Source: Adapted from G. Merriman's To Discover Mathematics, 1942, p. 80


Hint: First, try to form a definition of "exact" sum which makes sense...mathematically.

Now, apply that definition...

 


Solution Commentary: No definite answers provided, as "answers" will depend on the definition of "exact" that is used.

If you want to explore this "inexact" area more, explore the ideas proposed by Richard Dedekind around 1854.

Finally, as to the last question, it is important you work out the problem using a "large" number of digits in each number, as new patterns replace old patterns, etc.