Note. We use several algebraic properties of the real numbers to verify corresponding properties of complex numbers. As we should at this level, we give the results in a theorem/proof format.
Theorem:
\[For\;any\;\;{z_1},{z_2},{z_3} \in C\;\;\;\;,\;we\;have\;the\;following:\]1. Commutivity of addition and multiplication:
\[{z_1} + {z_2} = {z_2} + {z_1}\;\;\;\;\;\;,\;and\;\;\;{z_1}{z_2} = {z_2}{z_1}\;\]2. Associativity of addition and multiplication:
\[\left( {{z_1} + {z_2}} \right) + {z_3} = {z_1} + ({z_2} + {z_3})\;\;\;\;\;,\;and\;\left( {\;{z_1}{z_2}} \right){z_3} = {z_1}({z_2}{z_3})\]3. Distribution of multiplication over addition:
\[{z_1}({z_2} + {z_3}) = {z_1}{z_2} + {z_2}{z_3}\]4. There is an additive identity 0 = 0 + i0 such that 0 + z = z for all z ∈ C. There is a multiplicative identity 1 = 1 + i0 such that z1 = z for all z ∈ C. Also, z0 = 0 for all z ∈ C.
5. For each z ∈ C there is z' ∈ C such that z' + z = 0. z0 is the additive inverse of z (denoted -z). If z doesn't equal 0, then there is z'' ∈ C such that z'' z = 1. z'' is the multiplicative inverse of z (denoted z-1).
Note. When we consider division it is, by definition, multiplication by the multiplicative inverse.
\[\begin{array}{l}
So\;for\;z \ne 0\;,\;{z^{ - 1}}\;is\;denoted\;\frac{1}{z}\;and\;so\;\frac{{{z_1}}}{{{z_2}}}\;means\;{z_1}z_2^{ - 1}\;.\\
\;We\;see\;that\;for\;z = x + iy\;:\\
{z^{ - 1}} = \frac{1}{z} = \frac{x}{{{x^2} + {y^2}}} + i\frac{{ - y}}{{{x^2} + {y^2}}}
\end{array}\]
Tags:
Complex Numbers