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Thought-Process to Discover Knowledge

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mathsComplex NumbersProperties of Complex Number Arithmetic

Multiplication : Associative Law

Complex multiplication is associative : Order of multiplication can be changed.



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What does 'Associate' mean?Associative property illustration

  • to connect with; to join
  • to connect with; to join
  • to separate

The answer is 'to connect with; to join'

When three complex numbers are multiplies,
 •  Person A associates the middle complex number with first complex number and, to the result of that, he multiplies the third complex number `(z_1 xx z_2) xx z_3`
 •  Person B associates the middle complex number with third complex number and to that she multiplies the first complex number `z_1 xx (z_2 xx z_3)` Associative property illustration will the two persons get different results?

  • They will get different results.
  • They will get identical results
  • They will get identical results

The answer is 'They will get identical results'.

 •  Order of multiplication can be changed - associative

Associative property of Complex Multiplication: For Any complex number `z_1,z_2,z_3 in CC`,
`z_1 xx (z_2 xx z_3) = (z_1 xx z_2) xx z_3`

Is Associative property applicable to division?

  • when Division is handled as inverse of multiplication.
  • when Division is handled as inverse of multiplication.
  • Order of division can be swapped

The answer is 'when Division is handled as inverse of multiplication'. For `(z_1 -: z_2) -: z_3`, it will be handled `(z_1 xx 1/z_2)xx 1/z_3`, then associative property of multiplication applies.

Solved Exercise Problem:

Given `z_1, z_2 in CC`, what will be the result of `z_1 xx (z_2 xx z_1)`?

  • `z_1^2 xx z_2`
  • `z_1^2 xx z_2`
  • `z_1 xx z_2`
  • `z_2`

The answer is '`z_1^2 xx z_2`'.

                            
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