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Complex numbers have multiplicative inverse.



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What does 'inverse' mean?

  • same; identical; similar
  • opposite; converse
  • opposite; converse

The answer is 'opposite; converse'.

Consider two complex numbers `z_1 = a+ i b` and `z_2 = 1 /(a+ib)`. What will be `z_1 xx z_2`?

  • `1`
  • `1+0i`
  • multiplicative identity
  • All the above
  • All the above

The answer is 'All the above'

Consider two complex numbers `z_1 = a+ i b` and `z_2 = 1 /(a+ib)`.

`z_2 = 1/(a+ib)`
`quad quad quad = 1/(a+ib) xx (a-ib)/(a-ib) `
`quad quad quad = (a-ib)/(a^2+b^2)`

`1/z` or (a-ib)/(a^2+b^2) is the multiplicative inverse.

 •  For any complex number `z` there exists `1/z` : multiplicative inverse

Multiplicative Inverse: For any complex number `z = a+ib in CC`, there exists `1/z = (a - i b)/(a^2+b^2) in CC` such that
`z xx 1/z = 1`

Solved Exercise Problem:

What is the multiplicative inverse of `2-i`?

  • `2+i`
  • `(2+i)/5`
  • `(2+i)/5`
  • `(2+i)/sqrt(5)`

The answer is '`(2+i)/5`'

                            
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