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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

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Complex multiplication is closed : Product or quotient of two complex number is another complex number



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What does 'closure' mean?closure property illustration

  • closed
  • not open
  • both the above
  • both the above

Answer is 'both the above'

Given complex numbers `z_1=a_1+b_1i` and `z_2=a_2+b_2i`, `a_1,a_2,b_1,b_2 in R`, will `z_1 xx z_2` a complex number?

  • No. When multiplied, the result has `i^2`
  • Yes. When multiplied, the result consists of real numbers, as real and imaginary parts
  • Yes. When multiplied, the result consists of real numbers, as real and imaginary parts

The answer is 'Yes. When multiplied, the result consists of real numbers as real and imaginary parts'

 •  Product or quotient of two complex number is another complex number - closed

Closure property of Multiplication and Division: For any given complex numbers `z_1, z_2 in CC`, ` z_1 xx / -: z_2 in CC`.

Solved Exercise Problem:

Given complex numbers `z_1=a_1+b_1i` and `z_2=a_2+b_2i`, `a_1,a_2,b_1,b_2 in R`, will `z_1 -: z_2` a complex number?

  • No. Complex division is not defined
  • Yes. complex number division is equivalently multiplication by the conjugate
  • Yes. complex number division is equivalently multiplication by the conjugate

The answer is 'Yes. When multiplied, the result consists of real numbers as real and imaginary parts'

                            
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