Combined

Definition

Combined of a Complex number z = has + Bi \, , where has and B is real, is z' = has - Bi \, and it is often noted \ bar z, read “Z bars”.

In the plan, the point of affix \ bar z is symmetrical point of affix z \, compared to the x-axis.

The module of combined remains unchanged. The conjugation is a continuous linear operation.

Properties

  • \ overline {z+w} = \ bar Z + \ bar w
  • \ overline {zw} = \ bar Z \ times \ bar w
  • \ overline {\ left (\ frac {Z} {W} \ right)} = \ frac {\ bar Z} {\ bar W} if w \, is not-no one
  • if \ operatorname {Im} \ left (Z \ right) = 0 then \ bar Z = z
  • \ left|\ bar Z \ right| = \ left|Z \ right|
  • z \ overline {Z} = \ left|Z \ right|^2
  • z^ {- 1} = {\ overline {Z} \ over {\ left|Z \ right|^2}} for Z not-no one.

Quaternion S

Combined quaternion q = has + Bi + cj + dk \, is q^* = has - bi- cj - dk \, .

Property

  • q \ cdot q^* = a^2+b^2+c^2+d^2 \,

  • \ frac {1} {Q} = \ frac {1} {(a^2+b^2+c^2+d^2)}\ cdot q^* \,

  • One can easily calculate the reverse of a quaternion by using the properties of the quaternion combined

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