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notes on conjugates and modulus
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Conjugate And ModulusIn the previous section we looked at algebraic operations on complex numbers.Th
other operations that we should take a look at since they tend to show up on occa
a look at quite a few nice facts about these operations.
Complex Conjugate
The first one we’ll look at is the complex conjugate, (or just the conjugate).Given
number z = a + bi the complex conjugate is denoted by ¯z̄¯ and is defined to be,
¯z̄
¯ = a − bi
In other words, we just switch the sign on the imaginary part of the number.
Here are some basic facts about conjugates.
¯¯¯
¯z̄
¯ =z
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
z1 ± z2 = ¯z̄¯ 1 ± ¯z̄¯ 2
¯z¯¯¯¯¯z
1 2
¯¯¯ = ¯z̄¯ 1 ¯z̄¯ 2
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯ ¯z̄
¯
z1
( )=
1
z2 ¯z̄
¯
2
The first one just says that if we conjugate twice we get back to what we started w
hopefully this makes some sense.The remaining three just say we can break up s
products and quotients into the individual pieces and then conjugate.
So, just so we can say that we worked a number example or two let’s do a couple
illustrating the above facts.
Example 1 Compute each of the following.
¯¯¯
(a) ¯z̄¯ for z = 3 − 15i
¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯
(b) z1 − z2 for z1 = 5 + i and z2 = −8 + 3i
(c) ¯z¯¯¯1¯ − ¯z¯¯¯2¯ for z1 = 5 + i and z2 = −8 + 3i
Hide Solution !
There really isn’t much to do with these other than to so the work so,
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