Division of Imaginary Numbers
Dividing two imaginary numbers produces a real number. $$ (0,a) : (0,b) = (0,a:b) $$ When expressed in algebraic form, where (0,b)=bi and (0,a)=ai, the division becomes: $$ ai : bi = a : b $$ Here, \( i=(0,1) \) represents the imaginary unit.
The Proof
Let’s consider two imaginary numbers:
$$ z_1 = (0,a)=ai $$ $$ z_2 = (0,b)=bi $$
To divide these imaginary numbers in Cartesian form, (0,a) and (0,b), we use the rule for dividing complex numbers:
$$ z_1 : z_2 = (0,a) : (0,b) $$
$$ z_1 : z_2 = \frac{0 \cdot 0 + a \cdot b}{0^2+b^2} + \frac{0 \cdot a - 0 \cdot b}{0^2+b^2} \cdot i $$
$$ z_1 : z_2 = \frac{0 + ab}{b^2} + \frac{0 - 0}{b^2} \cdot i $$
$$ z_1 : z_2 = \frac{ab}{b^2} $$
By simplifying, dividing the numerator and denominator by \( b \):
$$ z_1 : z_2 = \frac{a}{b} $$
Therefore, the result is the real number \( a:b \).
When performing the division of imaginary numbers in algebraic form, we arrive at the same result:
$$ z_1 : z_2 = ai : bi $$
Rewriting the division as a fraction gives:
$$ z_1 : z_2 = \frac{ai}{bi} $$
By canceling the imaginary unit \( i \) in both the numerator and denominator:
$$ z_1 : z_2 = \frac{a}{b} $$
Note. Division in algebraic form is much simpler since it follows the basic rules of real number arithmetic. There’s no need to apply the complex number division formula.
A Practical Example
Let’s take two imaginary numbers:
$$ (0,6)=6i $$ $$ (0,3)=3i $$
The division of these two imaginary numbers results in the real number \( (2,0) \):
$$ (0,6) :(0,3) = \frac{0 \cdot 0 + 6 \cdot 3}{0^2+3^2} + \frac{0 \cdot 6 - 0 \cdot 3}{0^2+3^2} \cdot i $$
$$ (0,6) :(0,3) = \frac{0 + 18}{0+9} + \frac{0 - 0}{0+9} \cdot i $$
$$ (0,6) :(0,3) = \frac{18}{9} + 0 \cdot i $$
$$ (0,6) :(0,3) = 2 $$
Note. Writing \( 2 \) or \( (2,0) \) is equivalent because the imaginary part of the complex number is zero, leaving only the real part. This makes the result the real number \( 2 \).
Using algebraic form for the same numbers yields the same result:
$$ 6i : 3i = \frac{6i}{3i} $$
$$ 6i : 3i = \frac{6}{3} $$
$$ 6i : 3i = 2 $$
The result is always the real number \( (2,0)=2 \).
Note. When imaginary numbers are in Cartesian form \( (0,b) \), division must follow the complex number division formula: $$ (a,b) : (c,d) = \frac{ac+bd}{c^2+d^2} + \frac{cb-ad}{c^2+d^2} $$ On the other hand, when imaginary numbers are in algebraic form, a simple division of real numbers suffices because the imaginary unit cancels out: $$ ai : ci = \frac{ai}{ci} = \frac{a}{b} $$ Thus, division in algebraic form is significantly faster and more straightforward.
And so on.
