How to find the multiplicative inverse of imaginary numbers
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The multiplicative inverse of a complex number z=x+iyz=x+iy where x,yx,y are real is the number c=a+ibc=a+ib such that
z×c=c×z=1z×c=c×z=1
⇒ax−by=1⇒ax−by=1 and ay+bx=0ay+bx=0
⇒ax+ayxy=1⇒ax+ayxy=1
⇒a=xx2+y2⇒a=xx2+y2 and b=−yx2+y2b=−yx2+y2
Note that the conjugate of z=x+iyz=x+iy is written z¯=x−iyz¯=x−iy, so we have
z×z¯=x2+y2=|z|2z×z¯=x2+y2=|z|2
Hence
1z=z¯|z|21z=z¯|z|2
Conjugates are a natural symmetry in complex numbers that follows from the fact that, in some sense, it is impossible to distinguish +i+i and −i−ibecause they both have exactly the same properties.
In polar notation, z=reiθz=reiθ for real r≥0r≥0 and θ∈(−π,π]θ∈(−π,π]. The multiplicative inverse (for r≠0)r≠0) is
1z=(reiθ)−1=r−1e−iθ=e−iθr1z=(reiθ)−1=r−1e−iθ=e−iθr
There are lots of ways to look at it :-)
z×c=c×z=1z×c=c×z=1
⇒ax−by=1⇒ax−by=1 and ay+bx=0ay+bx=0
⇒ax+ayxy=1⇒ax+ayxy=1
⇒a=xx2+y2⇒a=xx2+y2 and b=−yx2+y2b=−yx2+y2
Note that the conjugate of z=x+iyz=x+iy is written z¯=x−iyz¯=x−iy, so we have
z×z¯=x2+y2=|z|2z×z¯=x2+y2=|z|2
Hence
1z=z¯|z|21z=z¯|z|2
Conjugates are a natural symmetry in complex numbers that follows from the fact that, in some sense, it is impossible to distinguish +i+i and −i−ibecause they both have exactly the same properties.
In polar notation, z=reiθz=reiθ for real r≥0r≥0 and θ∈(−π,π]θ∈(−π,π]. The multiplicative inverse (for r≠0)r≠0) is
1z=(reiθ)−1=r−1e−iθ=e−iθr1z=(reiθ)−1=r−1e−iθ=e−iθr
There are lots of ways to look at it :-)
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