Divide Complex Numbers Calculator

Dividing complex numbers depends primarily on the format of the inputs. In rectangular form, given two numbers $z_1=a_1+b_{1}i$ and $z_2=a_2+b_{2}i$, division is performed by multiplying both the numerator and the denominator by the complex conjugate of the denominator ($a_2-b_{2}i$). This yields the formula for the real part: $\frac{a_1{a}_2+b_1{b}_2}{a_2^2+b_2^2}$ and the imaginary part: $\frac{b_1{a}_2-a_1{b}_2}{a_2^2+b_2^2}$. In polar form ($r\angle \theta$), division is much simpler: you divide the magnitudes and subtract the denominator's phase angle from the numerator's phase angle: $\frac{r_1}{r_2}\angle ({\theta }_1-{\theta }_2)$.

Let's divide two complex numbers in rectangular form: $z_1=4+2i$ by $z_2=1-3i$.

Step 1: Identify components. Here, $a_1=4$, $b_1=2$, $a_2=1$, and $b_2=-3$.

Step 2: Find the denominator's squared magnitude. Calculate $a_2^2+b_2^2=1^2+(-3{)<!--</mo--><mn>2</mn>}^{}=1+9=10$.

Step 3: Compute the real part. Apply the formula $(a_1{a}_2+b_1{b}_2)/10$, which gives $\left(\right(4\times 1)+(2\times -3\left)\right)/10=(4-6)/10=-0.2$.

Step 4: Compute the imaginary part. Apply the formula $(b_1{a}_2-a_1{b}_2)/10$, resulting in $\left(\right(2\times 1)-(4\times -3\left)\right)/10=(2+12)/10=1.4$.

Final Answer: The quotient is $-0.2+1.4i$.

Convert to Polar for Multiple Multiplications/Divisions: If your equation involves a heavy chain of products and quotients, converting your rectangular numbers into polar form first will save you considerable time by replacing cumbersome binomial expansions with simple addition and subtraction of angles.

Beware of Division by Zero: Just like with real numbers, dividing by zero ($0+0i$) is undefined. Always double-check your denominator's magnitude before finalizing calculations.

Rationalize the Denominator: When doing homework by hand, always check your work by multiplying your final quotient back by the original divisor; you should arrive precisely back at your initial numerator.