Basic Algebraic Properties For Complex Numbers
Published On :2021-03-07 22:51:00
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}}} + ifrac{{ - y}}{{{x^2} + {y^2}}}
end{array}]