Why is ln(k) a Complex Number When k is a Positive Integer?

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SUMMARY

The discussion centers on the nature of the natural logarithm function, specifically why \(\ln(k)\) is considered a complex number when \(k\) is a positive integer. Participants clarify that the natural logarithm can be expressed in terms of complex numbers using Euler's formula, which states that \(\ln(k) = \ln|k| + i\arg(k)\). Since \(k\) is a positive integer, \(\arg(k)\) equals zero, leading to \(\ln(k) = \ln(k) + 0i\), thus confirming that \(\ln(k)\) is indeed a complex number with an imaginary part of zero.

PREREQUISITES
  • Understanding of complex numbers and their representation
  • Familiarity with the natural logarithm function
  • Knowledge of Euler's formula and its implications
  • Basic concepts of argument and modulus in complex analysis
NEXT STEPS
  • Study the properties of complex logarithms and their applications
  • Explore Euler's formula in depth, including its derivations and uses
  • Learn about the geometric interpretation of complex numbers
  • Investigate the implications of complex analysis in various mathematical fields
USEFUL FOR

Mathematicians, students of complex analysis, and anyone interested in the properties of logarithmic functions in the context of complex numbers.

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Why ln(k) when k is a possitive integer, ln(k) is a complex number?
 
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Why do you think \(\ln(k)\) is complex?
 

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