{"id":3999,"date":"2015-07-30t09:00:36","date_gmt":"2015-07-30t16:00:36","guid":{"rendered":"\/\/www.catharsisit.com\/act\/?p=916"},"modified":"2015-08-11t17:09:47","modified_gmt":"2015-08-12t00:09:47","slug":"complex-numbers-on-the-act-multiplication-and-division","status":"publish","type":"post","link":"\/\/www.catharsisit.com\/hs\/act\/complex-numbers-on-the-act-multiplication-and-division\/","title":{"rendered":"complex numbers on the act: multiplication and division"},"content":{"rendered":"

remember, a complex number<\/a> is very similar to a binomial. we\u2019re dealing with imaginary and real numbers at the same time. we already took a look at addition and subtraction, so let\u2019s move on to multiplication and division. these are a little trickier, but only division involves a skill you may not have used yet. take a look!<\/p>\n

 <\/p>\n

multiplication<\/h2>\n

do you remember having to learn how to foil? you multiply the terms of a binomial or complex number in this order: first, outer, inner, last. let\u2019s take a look at how to do it with a complex number.<\/p>\n

\"cnmad_img1\"<\/p>\n

that leaves us with this:<\/p>\n

\"cnmad_img2\"<\/p>\n

now remember, \"{i^2}, as we already covered. so we get this:<\/p>\n

\"cnmad_img3a\"<\/p>\n

division<\/h2>\n

all right, here\u2019s where things get a little tricky, but stick with me. i promise, we\u2019ll come out on the other side (mostly) unscathed.<\/p>\n

let\u2019s say you had to divide 5 + 2i<\/em> by 6 + 3i<\/em>.<\/p>\n

\"cnmad_img4\"<\/p>\n

now, remember, i<\/em> is just another way of writing \u221a-1. and, according to the ancient laws of math, we can\u2019t have a radical in the denominator (or bottom part) of a fraction. so, it looks like we have to simplify in order to solve this problem.<\/p>\n

step one: conjugate<\/em><\/p>\n

in order to divide complex numbers, what you have to do is multiply by the complex conjugate of the denominator.<\/strong> i heard about half of you get sudden migraines there, but i promise, that\u2019s not as complicated as it sounds. the complex conjugate<\/strong> is just the same exact denominator<\/em> with one tiny change. instead of 6 + 3i<\/em>, we take 6 – 3i.<\/em><\/p>\n

so our problem now looks like this:<\/p>\n

\"cnmad_img5\"<\/p>\n

really, all we\u2019re doing is multiplying by a fancy form of 1, so we\u2019re not actually changing<\/em> the problem; we\u2019re just simplifying it.<\/p>\n

step two: multiply<\/em><\/p>\n

it looks like we\u2019re out of plastic wrap, which is okay, because all we need is foil. yes, the good old first-outer-inner-last method of multiplying binomials and complex numbers is back again. and this time, it\u2019s personal.<\/em><\/p>\n

okay, not really. but let\u2019s foil anyway. we\u2019ll do the numerator first.<\/p>\n

\"cnmad_img6\"<\/p>\n

that leaves us with this:<\/p>\n

\"cnmad_img7\"<\/p>\n

and now, do the denominator the same way:<\/p>\n

\"cnmad_img8\"<\/p>\n

step three: simplify<\/em><\/p>\n

here\u2019s our problem so far:<\/p>\n

\"cnmad_img9\"<\/p>\n

we already know that\u00a0\"{i^2},\u00a0so let\u2019s change that in both the numerator and the denominator.<\/p>\n

\"cnmad_img10\"<\/p>\n

and now, combine like terms! watch the magic!<\/em><\/p>\n

\"\"<\/a><\/p>\n

notice how the denominator suddenly doesn\u2019t have any more i<\/em> in it. we\u2019ve fully simplified this problem! woo-hoo! take a nice deep breath, magooshers! you\u2019ve earned it.<\/p>\n

 <\/p>\n","protected":false},"excerpt":{"rendered":"

remember, a complex number is very similar to a binomial. we\u2019re dealing with imaginary and real numbers at the same time. we already took a look at addition and subtraction, so let\u2019s move on to multiplication and division. these are a little trickier, but only division involves a skill you may not have used yet. […]<\/p>\n","protected":false},"author":92,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[90],"tags":[9],"ppma_author":[24906],"acf":[],"yoast_head":"\ncomplex numbers on the act: multiplication and division - magoosh blog | high school<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"\/\/www.catharsisit.com\/hs\/act\/complex-numbers-on-the-act-multiplication-and-division\/\" \/>\n<meta property=\"og:locale\" content=\"en_us\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"complex numbers on the act: multiplication and division\" \/>\n<meta property=\"og:description\" content=\"remember, a complex number is very similar to a binomial. we\u2019re dealing with imaginary and real numbers at the same time. we already took a look at addition and subtraction, so let\u2019s move on to multiplication and division. these are a little trickier, but only division involves a skill you may not have used yet. 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