Sure, let's divide two complex numbers step by step.
We are given the complex numbers (3+1i) and (2-3i). Our goal is to divide these two complex numbers.
To find the quotient of two complex numbers, we multiply the numerator and the denominator by the conjugate of the denominator.
The conjugate of a complex number (a+bi) is the complex number (a-bi). So, the conjugate of (2-3i) is (2+3i).
Consider the division (3+1i) / (2-3i)
We start by multiplying both the numerator and the denominator by the conjugate of the denominator, which is (2+3i).
= ((3+1i)(2+3i)) / ((2-3i)(2+3i))
= (6+9i+2i+3) / ((4+6i-6i-9))
= (9+11i) / (4-9)
= (9+11i) / -5
We split the fraction into real and imaginary parts:
= 9/-5 + 11i/-5
= -9/5 - 11i/5
So, the result of the division (3+1i) / (2-3i) is -9/5 - 11i/5.