Hello wonderful mathematics people. I'm Anna Cox from Kella Community College. Imaginary number sqrt -1 equal I also -1 equal I ^2. Complex numbers are of the form A+ BI where A&B are real numbers. The conjugate of a complex number, also known as the complex complex conjugate of A+ BI is A minus BI. If I take and I multiply together the conjugates A+ BI times a minus BI using my foil, I would get a squared minus ABI plus ABI minus b ^2 I ^2. Well, if we combine like terms, the negative ABI and the positive ABI are going to cancel I ^2 down here is really -1. So negative b ^2 * -1 would give us a positive b ^2. So the complex conjugate's a very useful manner to get rid of I's. If we wanted to do a division problem, say we had 2 - 3 I over 4 + 5 I, we'd multiply by its conjugate and the conjugate would be 4 - 5 I. And if we do the bottom, we have to do the top 4 - 5 I. If we foil this out, we'd get 8 -, 10 I -12 I plus 15I squared on the top over 16 - 20 I plus twenty I -, 25 I squared. So on the top, this I ^2 turns into a negative 1/8 and -15 would give us -7 -, 22 I for the top portion numerator. On the bottom, the -20 I and +20 I cancel I ^2 turns into a -1, so we get 16 + 25, which is 41. If we were asked to actually write it as a complex number, it'd be -741 -, 2241. I thank you and have a wonderful day. This is Anna Cox.