Betsy McCall Courses -- Math Notes, Imaginary Numbers


homepage courses fall2004 spring2005 summer 2005 fall 2005 spring 2006 summer 2006 fall 2006 spring 2007 summer 2007 fall 2007 winter 2008 spring 2008 summer 2008 fall 2008 mathnotes resources previous -- view my teaching portfolio here		Math Notes Imaginary Numbers Introduction Before we introduce the notion of a complex number we must first introduce one of the elements of a complex number, an imaginary number. Imaginary numbers are numbers which are given by the or some multiple of this. By a multiple of this, we can mean anything like or we might mean (Remember that these two are the same, from properties of square roots: ). In both cases we have a negative under the square root sign. When we learned basic algebra, we learned that these numbers have no value in the real world. There are no real numbers that we can multiply together to get any negative number. When we have encountered these numbers in the past, we simply discarded them as meaningless. What we are going to do now with imaginary numbers, and then with complex numbers in pretend they exist and see where this gets us. It's amazing how much of math was originally done like this... even negative numbers and zero were mere fantasy until someone came up with a use for them. And as one can see if one ever takes physics, engineering or any higher level math courses, there are a lot of ways that imaginary and complex numbers turn out to be both useful and meaningful. In order to help us deal with these imaginary numbers, we are going to first get rid of the square roots. Whenever we find a negative under the square root, we can always separate out the factor of as we did in the example above. Then we are going to replace that with the letter i. This letter i stands for the imaginary number, the . So that now can be written as 2i. (It should be noted here that in some disciplines, other characters are used. In engineering for instance, it's common to represent the imaginary number with a j, because i is used for current. However, we will use i throughout our discussion here.) Because i is defined to be , i also has another important property: i² = -1. We can use this property to determine other powers of i, as well as the value one obtains when multiplying numbers containing i. Worked Examples A. Rewrite the following number as an imaginary number containing i. Reduce the square roots as much as possible. Remember, first factor out the , giving us . Note that the i is outside the square root. B. Add the following imaginary numbers: 2i and 3i. Well, one adds 2i + 3i just like one adds 2x + 3x. They are like terms, so add the coefficients. 2i + 3i = (2 + 3)i = 5i. C. What is i³? i⁴? i¹⁷? i⁹⁵? i³ = i²i = (-1)i = -i. Use the definition of i² = -1 to reduce this problem. i⁴ = i²i² = (-1)(-1) = 1. Again, we used the definition of i² = -1. We'll use this property in the next example, too. i¹⁷ = i¹⁶i = (i⁴)⁴i = (1)⁴* i = i. We have an odd number, so separate out one of the i's to make one of the factors even. See how the exponent divides by 4? We can use here the property from the previous example, that i⁴ = 1 to reduce the problem further. i⁹⁵ = i⁹⁴i = (i²)⁴⁷i = (-1)⁴⁷i = (-1)i = -i. As with the previous problem, we have an odd exponent, so we factor out one of the i's and attempt to reduce what remains. (Notice that we are only reducing using even powers.) The exponent 94 isn't divisible by 4, so we will use the original definition to reduce this term to a power of i². (-1) to any odd power is still -1. Notice that powers of i go through a cycle, i, -1, -i, 1, i, -1, -i, 1, ... (each number in this list is i, i², i³, i⁴, i⁵, ...) D. Find the value of (3i)(4i). Just as though we were multiplying 3x4x, we will multiply the coefficients and the 'variables'. So 3i4i = (34)(ii) = 12i² = 12(-1) = -12. That last step used the definition i² = -1. Problem Solving Tips* If you feel like dealing with imaginary numbers is confusing because you don't understand what the really is, try not to think about it, and instead try to think of i as you would a variable, it's an unknown value, but it works pretty much the same as dealing with an x. When adding imaginary numbers you add them like as you would like terms, add the coefficients. When multiplying imaginary numbers, multiply them as one would with variables... multiply the coefficients and add the exponents of the variable. When reducing powers of i, if it's even, remember, you can divide the exponent by two, and then raise (-1) to that new power using the definition of i. If the exponent is odd, factor out an i, and then reduce the factor with the now even exponent. Make sure that your final answer has no power of i greater than 1. Additional Problems with Answers 1. Rewrite the following as imaginary numbers. a. b. c. d. e. f. g. 2. Add the following imaginary numbers. a. i + 3i b. 2i - 4i c. d. 3. Multiply the following imaginary numbers. a. i4i b. (-i)(-6i) c. i²i⁵ d. 4. Simplify the following imaginary numbers. a. (i + 2i)(3i) b. (-i - 7i)(i²) c. (4i)(i² - 3i) 5. Reduce the following imaginary numbers as much as possible, a. i6 b. i13 c. i23 d. i56 e. i81 f. i107 g. i4025 Answers 1. a. 3i, b. 8i, c. , d. , e. , f. , g. 2. a. 4i, b. -2i, c. , d. 3. a. -4, b. -6, c. -i, d. -1/2 4. a. -9, b. 8i, c. -4i -12 5. a. -1, b. i, c. -i, d. 1, e. i, f. -i, g. i Links to Outside Sources Math Forum: Ask Dr. Math FAQ: Imaginary Numbers Answers and Explanations -- Do "Imaginary Numbers" Really Exist? Imaginary Number -- from MathWorld Imaginary Numbers Imaginary Numbers Imaginary Numbers Are Not Imaginary. Imaginary Numbers imaginary number@Everything2.com Imaginary Numbers are not Real - the Geometric Algebra of Spacetime imaginary number: Definition and Much More From Answers.com Links to Supporting Topics Complex Numbers in Cartesian Form (a + bi) Complex Numbers in Polar or Trigonometric Form and Exponential Form (rcos(t) + risin(t)) & re^it Definition of Complex and Imaginary Numbers Complex numbers: real and imaginary parts Math Ally - Imaginary and complex numbers What's a number? Lesson on Complex Numbers or Imaginary Numbers History Question Corner -- The Origin of Complex Numbers A History of Hypercomplex Numbers Re: so, who invented the IMAGINARY NUMBER? Wikipedia article

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