{"id":3999,"date":"2015-07-30T09:00:36","date_gmt":"2015-07-30T16:00:36","guid":{"rendered":"https:\/\/magoosh.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":"https:\/\/magoosh.com\/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],"class_list":["post-3999","post","type-post","status-publish","format-standard","hentry","category-all","tag-act-math"],"acf":[],"yoast_head":"\nComplex Numbers on the ACT: Multiplication and Division - Magoosh Blog | ACT<\/title>\n<meta name=\"description\" content=\"Remember, a complex number is very similar to a binomial. We\u2019re dealing with imaginary and real numbers at the same time. 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