Proving Complex Numbers are Rational

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SUMMARY

The discussion centers on proving that if the magnitudes |Z| and |W| of complex numbers Z and W are rational, and the difference W - Z is also rational, then the expression (1/Z) - (1/W) is rational. Participants clarify that it is the magnitudes of the complex numbers that are rational, not the numbers themselves. The distinction between the notation |Z| and /Z/ is emphasized, highlighting the importance of accurate representation in mathematical expressions.

PREREQUISITES
  • Understanding of complex numbers and their properties
  • Familiarity with rational numbers and their representation
  • Knowledge of basic algebraic manipulation
  • Ability to interpret mathematical notation accurately
NEXT STEPS
  • Study the properties of complex numbers in depth
  • Learn about rational and irrational numbers in mathematical contexts
  • Explore algebraic manipulation techniques involving complex numbers
  • Investigate proofs involving complex number operations and their implications
USEFUL FOR

Students studying complex analysis, mathematicians interested in number theory, and educators teaching advanced algebra concepts.

halvizo1031
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Homework Statement



Let Z and W be complex numbers. If /Z/ and /W/ are rational and /W-Z/ is rational, then
/(1/Z)-(1/W)/ is rational.

Homework Equations





The Attempt at a Solution


How do I represent Z and W as rational complex numbers?
 
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z and w are arbitrary complex numbers. Rational numbers can be written in the form a/b, where a and b are integers.

And it's |z|, not /z/.
 
halvizo1031 said:
How do I represent Z and W as rational complex numbers?

It isn't Z and W that are rational, it is their magnitudes |Z| and |W|.
 

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