How To Add Complex Fractions With Imaginary Numbers

in this video we're going to have a look at an example of adding and subtracting complex numbers sohere specifically we're given three different complex numbers a which is 3 minus 2i, bwhich is negative 7 plus i and c which is negative 5 minus 4i and we're asked to find

How To Add Complex Fractions With Imaginary Numbers, 3a minus 2b plus 9c first of all go ahead and try this problem on your own and then we'll take it up together here's what you're going to have to do you'regoing to have to make sure that you multiply

each of the coefficients so the 3, the negative 2, the 9 by the given complex number next you're going to have to add and subtract thecomplex numbers as indicated to do this remember that yousimply add or subtract the real parts and you also add or subtract theimaginary parts so just work with the real and the imaginary parts separately go ahead try this out and we'll continueonce you're done so hopefully you've had a chance to trythe out, let's go ahead and have a look at the solution so what we are trying to find

is 3a minus 2b plus 9c so lets substitute in what we know weknow that a is 3 minus 2i we know that b is negative 7 plus i and we know that c is negative 5 minus 4i now the first step is that we'regoing to have to you perform the indicated multiplication by thescalar the number outside by the given complex number so the 3 in the first term is gonna have tomultiply the 3 and the negative

2i the 2 in the second term is goingto have to multiply the negative 7 and the i and the 9 in the lastterm is going to have to multiply the negative 5 and the negative 4i so lets see what weget 3 times 3 gives us 9, 3 times negative 2igives us negative 6i the negative 2 times the negative 7 gives us positive 14 the negative 2 times the i gives us minus 2i the nine times the negative 5 gives us negative 45 and the nine times the negative 4i gives us minus 36i

so now our next step in order toactually simplifiy this is to group like terms in here like terms are real terms and imaginary terms so those of the two different sets that let's just go ahead and underlinein different colors the real terms and the imaginary terms so we can keep themstraight so right now let me just underline inred the real terms so the 9, the 14 and the negative 45 and in another color so let's choose blue here let me underlinethe imaginary terms

so negative 6i, negative 2i and negative 36i so first of all lets go ahead and addour real terms so we're going to have 9 plus 14 minus 45 and i'm just going to put them in brackets youdon't need to do this i'm just showing an extra step and now for the imaginary terms i've got negative 6i minus 2i minus 36i again we don't need thebrackets but they're just here to show us which termsare the real terms which terms the imaginary terms

so now if we go ahead and simplify9 plus 14 minus 45 the real terms that gives us negative 22 and minus 6i minus 22i minus 36i gives us negative 44i so altogether we have negative 22 that's the real component and negative 44i that's the imaginary component sooverall our complex number is negative 22 minus 44i