Sunday, June 28, 2020
Complex Numbers on the ACT Multiplication and Division
Remember, a complex number is very similar to a binomial. Weââ¬â¢re dealing with imaginary and real numbers at the same time. We already took a look at addition and subtraction, so letââ¬â¢s 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! Multiplication 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. Letââ¬â¢s take a look at how to do it with a complex number. That leaves us with this: Now remember, , as we already covered. So we get this: Division All right, hereââ¬â¢s where things get a little tricky, but stick with me. I promise, weââ¬â¢ll come out on the other side (mostly) unscathed. Letââ¬â¢s say you had to divide 5 + 2i by 6 + 3i. Now, remember, i is just another way of writing âËÅ¡-1. And, according to the ancient laws of math, we canââ¬â¢t 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. Step One: Conjugate In order to divide complex numbers, what you have to do is multiply by the complex conjugate of the denominator. I heard about half of you get sudden migraines there, but I promise, thatââ¬â¢s not as complicated as it sounds. The complex conjugate is just the same exact denominator with one tiny change. Instead of 6 + 3i, we take 6 3i. So our problem now looks like this: Really, all weââ¬â¢re doing is multiplying by a fancy form of 1, so weââ¬â¢re not actually changing the problem; weââ¬â¢re just simplifying it. Step Two: Multiply It looks like weââ¬â¢re 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, itââ¬â¢s personal. Okay, not really. But letââ¬â¢s FOIL anyway. Weââ¬â¢ll do the numerator first. That leaves us with this: And now, do the denominator the same way: Step Three: Simplify Hereââ¬â¢s our problem so far: We already know thatà ,à so letââ¬â¢s change that in both the numerator and the denominator. And now, combine like terms! Watch the magic! Notice how the denominator suddenly doesnââ¬â¢t have any more i in it. Weââ¬â¢ve fully simplified this problem! Woo-hoo! Take a nice deep breath, Magooshers! Youââ¬â¢ve earned it.
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