Imaginary Concepts -- Complex Numbers 1.1 2008/10/15 12:40:57.567 GMT-5 2008/11/15 13:36:31.643 US/Central Kenny Felder KFelder@RaleighCharterHS.org Kenny Felder KFelder@RaleighCharterHS.org algebra complex complex number i imaginary imaginary number This module introduces the concept of complex numbers in Algebra. A “complex number” is the sum of two parts: a real number by itself, and a real number multiplied by i. It can therefore be written as a+bi, where a and bi are real numbers. The first part, a, is referred to as the real part. The second part, bi, is referred to as the imaginary part. Examples of complex numbers a+bi (a is the “real part”; bi is the “imaginary part”) 3+2i a=3, b=2 π a=π, b=0(no imaginary part: a “pure real number”) -i a=0, b=-1 (no real part: a “pure imaginary number”) Some numbers are not obviously in the form a+bi. However, any number can be put in this form. Example 1: Putting a fraction into a+bi form (i in the numerator) 3−4i5 size 12{ { {3 - 4i} over {5} } } {} is a valid complex number. But it is not in the form a+bi, and we cannot immediately see what the real and imaginary parts are. To see the parts, we rewrite it like this: 3−4i5 size 12{ { {3 - 4i} over {5} } } {}= 35 size 12{ { {3} over {5} } } {}– 45 size 12{ { {4} over {5} } } {}i Why does that work? It’s just the ordinary rules of fractions, applied backward. (Try multiplying and then subtracting on the right to confirm this.) But now we have a form we can use: 3−4i5 size 12{ { {3 - 4i} over {5} } } {} a= 35 size 12{ { {3} over {5} } } {}, b=– 45 size 12{ { {4} over {5} } } {} So we see that fractions are very easy to break up, if the i is in the numerator. An i in the denominator is a bit trickier to deal with. Example 2: Putting a fraction into a+bi form (i in the denominator) 1i size 12{ { {1} over {i} } } {} = 1⋅ii⋅i size 12{ { {1 cdot i} over {i cdot i} } } {} Multiplying the top and bottom of a fraction by the same number never changes the value of the fraction: it just rewrites it in a different form. = i−1 size 12{ { {i} over { - 1} } } {} Because i•i is i2, or –1. =-i This is not a property of i, but of –1. Similarly,5−1 size 12{ { {5} over { - 1} } } {} = –5. 1i size 12{ { {1} over {i} } } {}: a=0, b=-1 since we rewrote it as -i, or 0-1i Finally, what if the denominator is a more complicated complex number? The trick in this case is similar to the trick we used for rationalizing the denominator: we multiply by a quantity known as the complex conjugate of the denominator. Definition of Complex ConjugateThe complex conjugate of the number a+bi is a-bi. In words, you leave the real part alone, and change the sign of the imaginary part. Here is how we can use the “complex conjugate” to simplify a fraction. Example: Using the Complex Conjugate to put a fraction into a+bi form 53−4i size 12{ { {5} over {3 - 4i} } } {} The fraction: a complex number not currently in the form a + b i = 5(3+4i)(3−4i)(3+4i) size 12{ { {5` \( 3+4i \) } over { \( 3 - 4i \) \( 3+4i \) } } } {} Multiply the top and bottom by the complex conjugate of the denominator = 15+20i32−(4i)2 size 12{ { {"15"+"20"i} over {3 rSup { size 8{2} } - \( 4i \) rSup { size 8{2} } } } } {} Remember, (x+a)(x–a)=x2–a2 = 15+20i9+16 size 12{ { {"15"+"20"i} over {9+"16"} } } {} (4i)2=42i2=16(–1)=–16, which we are subtracting from 9 = 15+20i25 size 12{ { {"15"+"20"i} over {"25"} } } {} Success! The top has i, but the bottom doesn’t. This is easy to deal with. = 1525 size 12{ { {"15"} over {"25"} } } {}+ 20i25 size 12{ { {"20"i} over {"25"} } } {} Break the fraction up, just as we did in a previous example. = 35 size 12{ { {3} over {5} } } {}+ 45 size 12{ { {4} over {5} } } {}i So we’re there! a= 35 size 12{ { {3} over {5} } } {} and b=45 size 12{ { {4} over {5} } } {} Any number of any kind can be written as a+bi. The above examples show how to rewrite fractions in this form. In the text, you go through a worksheet designed to rewrite −13 size 12{ nroot { size 8{3} } { - 1} } {} as three different complex numbers. Once you understand this exercise, you can rewrite other radicals, such as i size 12{ sqrt {i} } {}, in a+bi form.