Page 207 - 'Blast_Into_Math

Blast into Math! Analatic nummer theora: ants, ghosts and giants

• Hint for # 6: You can do this problem using only the definition of the set of complex
numbers together with the definition of i , and the properties of the set of real numbers R .
Remember that i = −1. For a non-zero complex number
2
z = a + ib,

either a =0 or b =0 or both a and b are not zero. The complex conjugate of z is written
¯ z and is the complex number

a − ib.

Show that when you multiply z and ¯z

2
2
z¯z = a + b ∈ R.
Since a and b cannot both be zero,

2
2
z¯z = a + b ∈ R.
Since R has multiplicative inverses for all non-zero real numbers

1
∈ R.
a + b 2
2

Now think about the complex number

a b
+ i .
2
2
a + b 2 a + b 2

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