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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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