### The Complex Conjugate Of 1I Is 1I In General To

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# The complex conjugate of (1+i) is (1−i). In general to obtain the complex conjugate reverse the sign of the imaginary part. (Geometrically this corresponds to finding the "mirror image" point…

The complex conjugate of (1+i) is (1−i). In general to obtain

the complex conjugate reverse the sign of the imaginary part.

(Geometrically this corresponds to finding the "mirror image" point

in the complex plane by reflecting through the x-axis. The complex

conjugate of a complex number z is written with a bar over it: z⎯⎯

and read as "z bar".

Notice that if z=a+ib, then

(z)(z⎯⎯)=|z|2=a2+b2

which is also the square of the distance of the point z from the

origin. (Plot z as a point in the "complex" plane in order to see

this.)

If z=1+3i then z⎯⎯ = and |z| = .

You can use this to simplify complex fractions. Multiply the

numerator and denominator by the complex conjugate of the

denominator to make the denominator real.

1+3i1−i= +i .

Two convenient functions to know about pick out the real and

imaginary parts of a complex number.

Re(a+ib)=a (the real part (coordinate) of the complex number),

and

Im(a+ib)=b (the imaginary part (coordinate) of the complex

number. Re and Im are linear functions — now that you

know about linear behavior you may start noticing it often.