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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.

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mathsComplex NumbersProperties of Complex Number Arithmetic

Multiplication : Commutative Law

Complex multiplication is commutative : Complex numbers can be swapped in complex multiplication.



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What does 'commute' mean?commutative property illustration

  • to go to and fro on a regular basis
  • to go to and fro on a regular basis
  • It is not an English word.

The answer is 'to go to and fro between two places on a regular basis'.

Consider the multiplication and division of complex numbers. `z_3 = z_1 xx z_2` given as `a_3+ib_3 = (a_1a_2-b_1b_2)+i(a_2b_1+a_1b_2)`, where `a_1, a_2, b_1, b_2 in RR`.

Will `z_1 xx z_2 = z_2 xx z_1`?

  • Yes. If one interchanges `a_1` and `a_2` ; `b_1` and `b_2`, the result is the same
  • Yes. If one interchanges `a_1` and `a_2` ; `b_1` and `b_2`, the result is the same
  • No. If one interchanges `a_1` and `a_2` ; `b_1` and `b_2`, the result is different.

The answer is 'Yes. If I interchange `a_1` and `a_2` ; `b_1` and `b_2`, the result is same'

 •  Complex numbers can be swapped in complex multiplication - commutative

Is commutative property defined for complex division?

  • No. Commutative property is not defined for inverse operations.
  • No. Commutative property is not defined for inverse operations.
  • Yes. Commutative property is defined for division

The answer is 'No. Commutative property is not defined for inverse operations.' Division is the inverse of multiplication.

Division is the inverse of multiplication.

The commutative property has to be used in the following way.
`z_1-:z_2`
`quad quad = z_1 xx (1/z_2)`
`quad quad =(1/z_2) xx z_1 (commutative property of multiplication)`

Commutative Property of Complex Multiplication: for any complex number `z_1, z_2 in CC`
`z_1 xx z_2 = z_2 xx z_1`

Solved Exercise Problem:

Given for two complex numbers `z_1, z_2`; `z_1 xx z_2 = 3.1+i2.9`. Find the value of `z_2 xx z_1`.

  • `3.1+i2.9`
  • `3.1+i2.9`
  • `3.1-i2.9`
  • `-3.1+i2.9`
  • `2.9+i3.1`

The answer is '`3.1+i2.9`'

                            
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