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For some practice working with complex numbers: Calculate (2 + 2i) + (1 – 2i) = (2 + 2i) – (1 – 2i) = (2 + 2i)(1 – 2i) = 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 = 2 + 2i 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. (2 + 2i) / (1 - 2i) = ? ? + 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.

          For some practice working with complex numbers:

Calculate
(2 + 2i) + (1 – 2i) =
(2 + 2i) – (1 – 2i) =
(2 + 2i)(1 – 2i) =

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 = 2 + 2i 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.

(2 + 2i) / (1 - 2i) = ? ? + 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.

$For some practice working with complex numbers: Calculate (2 + 2i) + (1 – 2i) = (2 + 2i) – (1 – 2i) = (2 + 2i)(1 – 2i) = The complex conjugate of (1 + i) is (1 – i). In general to obtain the complex conjugate reverse the sign of the imaginary part. (Geometrically this corresponds to finding the "mirror image" point in the complex plane by reflecting through the x-axis. The complex conjugate of a complex number z is written with a bar over it: z? and read as "z bar". Notice that if z = a + ib, then (z) (z?) = |z|² = a² + b² which is also the square of the distance of the point z from the origin. (Plot z as a point in the "complex" plane in order to see this.) If z = 2 + 2i then z? = and |z| = You can use this to simplify complex fractions. Multiply the numerator and denominator by the complex conjugate of the denominator to make the denominator real. (2 + 2i) / (1 - 2i) = ? ? + i ? ? Two convenient functions to know about pick out the real and imaginary parts of a complex number. Re(a + ib) = a (the real part (coordinate) of the complex number), and Im(a + ib) = b (the imaginary part (coordinate) of the complex number. Re and Im are linear functions – now that you know about linear behavior you may start noticing it often.$

Added by Michael R.

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