How do I eliminating complex numbers from the denominator?

In simplifying the complex expression 2+3i/5i and eliminating the complex number from the denominator, why is the answer positive? 3-2i/5


as opposed to -3+2i/-5? How do I eliminating complex numbers from the denominator?
multiply both numerator and denominator by i and you will get your answerHow do I eliminating complex numbers from the denominator?
Those who answered this question did not study it closely enough, alas.





It does not ask ';How do I eliminate UNREAL numbers from the denominator';, but speaks of COMPLEX numbers. The example given, however, is of an unreal denominator.





In the context of an unreal denominator, both answers are good.





However, suppose we have 1+2i as the denominator, in place of the 5i. If we write ';1-2i';, we have the ';complex conjugate';. We multiply top and bottom by the complex conjugate. Therefore, we have multiplied the expression by 1.





Now the denominator becomes (1+2i)(1-2i), which is 1-4. This is a real number, so there is nothing unreal in the denominator anymore, and the problem is easily solved.





If the result has both the real and the unreal negative, starting with a negative number is unavoidable, and is not wrong.





Vikram P said ';Multiply top and bottom by i';. In the case of an unreal denominator, this is exactly right. 3i on the top becomes 3 i-squared. i-squared is NEGATIVE.
In this case, you can multiply the denominator by any product of a constant and i. The easiest is of course i so multiplying numerator and denominator by i





[(2+3i)*i]/(5i)*i





(2+3i)* i = 2i + 3i*i = 2i +3(-1) = -3 + 2i





5i*i = 5i^2 = -5





So the fraction is





(-3 + 2i)/-5 = (3 - 2i)/5






by rationalizing the denominator we eliminate the complex number in the following way





(2 + 3i)/5i x - 5i /-5i (multiplying and dividing by its conjugate)





= (3-2i)/5 which is the required answer.








It is not (-3+2i)/-5 because this results comes when you multiply and divide with 5i which is not a conjugate of the given denominator.






Technically those are the same, but there's this rule that says you can't start with a negative so just to change the sign of everything.


3-2i is the same as -3+2i