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I appreciate this opportunity to submit a query on this forum.

When studying the continuous-time & discrete-time distinction, specifically with reference to discrete signals being identical when separated by 2*pi, it has struck me that a basic premise, per google calc, doesn't hold:

  • exp(2 * pi * sqrt(-1)) = 1
  • exp(4 * pi * sqrt(-1)) = 1
  • exp(6 * pi * sqrt(-1)) = 1
  • exp(8 * pi * sqrt(-1)) = 1

but...

  • exp(10 * pi * sqrt(-1)) = 1 - 1.2246468 × 10^-15 i

Any thoughts on the above (discrepancy?) would be appreciated.

Regards,

wirefree
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    $\begingroup$ This has nothing to do with discrete signals but with the computer representation of numbers. $\endgroup$ – Yves Daoust Sep 16 '15 at 12:47
  • $\begingroup$ pi does not equal $\pi$ exactly; it is an approximation (correct to $7$ or $8$ decimal places) to $\pi$ $\endgroup$ – Dilip Sarwate Sep 16 '15 at 12:56
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    $\begingroup$ If you use a program capable of symbolic (not just numeric) computation, you'll get the exact result. Example using Wolfram Alpha. $\endgroup$ – MBaz Sep 16 '15 at 13:27
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Writing another way

$$1 - 1.2246468 × 10^{-15} i = 1 - 0.0000000000000012246468i$$

It is just rounding errors adding up due to having a limited number of bits in whatever computer is doing the calculation.

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