Quandaries
and Queries 

Hi, My name is Josh, and I am a recent high school graduate who will take Calculus as a college freshman if he can pass the SAT II, but the question which someone gave you (not the one on logs, but the (a+b) + 5i = 9 + ai question) gave me trouble. Your solution was very helpful, so thank you for putting it online, but I have a problem with it. ai + b = ci + d would make it necessary that ai equal ci and b equal d, so a + b = 9, and ai = ci, or a = c, meaning that a = 5, so 5 + b = 9, and b = 4. However, if b is plugged into the original equation, the result is (a + 4) + 5i = 9 + ai, which would simplify to a + 5i = 5 + ai, for 4 is subtracted from each side, and it can be written as 5i  ai = 5  a, or i (5  a) = 5  a. (5  a) is (5  a) 1, and, using algebra, if it equals i (5  a), then i = 1, which is wrong. However, I was really impressed by your solution, and I'd like to know more tricks, just out of interest. (I call the one that you used "standard form analysis" because it deals with a problem in standard form; namely, ai + b = ci + d.) The reason why I brought up the plugin is because, besides a lot of tedious algebra, which can never get rid of the a in order to find b, plugging in answers was my only method, and 4, in addition to 5 and 9, was the only real multiple choice answer (the others were complex, but b has to be real). Thanks again, and please tell me about those other methods, Josh 

Hi Josh, Your solution to the problem of finding a and b when (a+b) + 5i = 9 + ai is good. You set the real part on the left equal to the real part on the right
and the imiginary part on the left equal to the imaginary part on the right
and then solved for a and b to get a=5 and b=4. You then tried substituting b=4 into the original equation to get
which you then rewrote as
At this point you divided both sides by 5  a to get i=1, which is not correct. So what went wrong? The problem is in the division in the line above. Whenever you simplify an algebraic expression by dividing both sides by the same amount you need to ensure that you are not dividing by zero. That is exactly the problem here. If a = 5 then 5  a = 0 and you are dividing by zero. Penny 

