Multiplying complex numbers

we're asked to multiply the complex number one minus 3i times the complex number two plus 5i and the general idea here is you can multiply these complex numbers like you would have as like you would have multiplied any traditional binomial you just have to remember that this isn't a variable this is the imaginary unit I or it's just I but we could do that in two ways we could just do the distributive property twice which I like a little bit more just because it actually you're doing it from a fundamental principle it's nothing new or you could use foil which you also use when you multiple you first multiply binomials and I'll do it both ways so you could view you could this is just a number one minus 3i and so we can distribute it over the two numbers inside of this expression so when we're multiplying it times this entire expression we can multiply 1 minus 3i times 2 and 1 minus 3i times 5i so let's do that so this can be re-written as 1 minus 3i times 2 times 2 I'll write the 2 out front plus 1 minus 3i plus 1 minus 3i times 5i times 5i all I did is the distributive property here all I said is look if I have a times B plus C this is the same thing as a B plus AC I just distributed I just distributed the a on the B or the C I distributed the 1 minus 3i on the 2 and the 5 I and then I can do it again I have a 2 now times 1 minus 3i I can distribute it 2 times 2 times 1 is 2 2 times negative 3i is negative 6i and over here I'll do it again 5i times 1 so it's plus 5i times 1 is 5 I and then 5i times negative 3i so let's be careful here 5 times negative 3 so let me just 5 times negative 3 is negative 15 negative 15 and then I have an I times and I write I'm multiplying five let me do this over here 5i times negative 3i that's the same thing as 5 times negative 3 times I times I so the five times negative three is negative 15 and then we have I times I which is I I squared now we know what I squared is by definition I squared is negative 1 I squared by definition is negative 1 so you have negative 15 times negative 1 well that's the same thing as positive 15 so this can be rewritten as 2 minus 6i plus 5i negative 15 times negative 1 is positive 15 now we can add the real parts we have a 2 and we have a positive 15 so 2 plus 15 and we can add the imaginary parts we have a negative 6 so we have a negative 6 or negative 6i I should say and then we have plus 5i so plus 5i + 2 + 15 is 17 and if I have negative 6 of something plus 5 of that something what do I have if I have 5 that's something and I take 6 of that something away then I have negative 1 of that something negative 6i plus 5i is negative 1 I or I could just say minus I so in this way I just multiplied these two expressions or these two complex numbers really I multiplied them just using the distributive property twice you could also do it using foil and I'll do that right now really fast it is a little bit faster but it's a little bit mechanical so you might forget why you're doing it in the first place but at the end of the day you are doing the same thing here you're essentially multiplying every term of this first number or every part of this first number times every part of the second number and foil just make sure that we're doing it and let me just write foil out here which I'm not a huge fan of but I'll do it just in case that's the way you're learning it so foil says let's do the first numbers let's multiply the first numbers so that's going to be the 1 times the 2 so 1 times the 2 that is the F in foil then it says let's multiply the outer numbers times each other so that's 1 times 5i so plus 1 times 5i this is the eoghan foil the outer numbers then we do the inner numbers negative 3i times 2 so the is negative 3i times 2 this is those are the inner numbers and then we do the last numbers negative 3i times 5 is so negative 3i times 5i these are the last numbers and that's all that foil is telling us it's just making sure we're multiplying every part of this number times every part of that number and then when we simplify it 1 times 2 is 2 1 times 5i is 5i negative 3i times 2 is negative 6i and negative 3i times 5i well we already figured with that we already figured out what that was negative 3i times 5i turns out to be 15 negative 3 times 5 is negative 15 but I times I is negative 1 negative 15 times negative 1 is positive positive 15 add to the real parts 2 plus 15 you get 17 added the imaginary parts you have 5i minus 6i you get negative I and once again you get the exact same answer