r 2x4 5x3 13x2 7x 5 hint x 1 2i is a zero 6

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Unformatted text preview: (−4) = i −4 = i(2i) = 2i2 = 2(−1) = −2, which is unacceptable.3 We are now in the position to define the complex numbers. Definition 3.5. A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit. √ 2 Complex numbers include things you’d normally expect, like 3 + 2i and 5 − i 3. However, don’t forget that a or b could be zero, which means numbers like 3i and 6 are also complex numbers. In √ other words, don’t forget that the complex numbers include the real numbers, so 0 and π − 21 are both considered complex numbers. The arithmetic of complex numbers is as you would expect. The only thing you need to remember are the two properties in Definition 3.4. The next example should help recall how these animals behave. Example 3.4.1. Perform the indicated operations and simplify. Write your final answer in the form4 a + bi. 1 Some technical mathematics textbooks label it ‘j ’. Note the use of the indefinite article ‘a’. Whatever beast is chosen to be i, −i is the other square root of −1. 3 We want to enlarge the number system so we can solve things like x2 = −1, but not at the cost of the established rules already set in place. For that reason, the general properties of radicals simply do not apply for even roots of negative quantities. 4 We’ll accept an answer of say 3 − 2i, although, technically, we should write this as 3 + (−2)i. Even we pedants have our limits. 2 220 Polynom...
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