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Unformatted text preview: 1 = 0 has no solution x ∈ R. In order to “solve” this equation, we have to introduce extra numbers into our number system.
Chapter 1 : The Number System page 6 of 19 First Year Calculus c W W L Chen, 1982, 2005 Deﬁne the number i by i2 + 1 = 0. We then extend the ﬁeld of all real numbers by adjoining the number i, which is then combined with the real numbers by the operations addition and multiplication in accordance with the Field axioms in Section 1.1. The numbers a + bi, where a, b ∈ R, of the extended ﬁeld are then added and multiplied in accordance with the Field axioms, suitably extended, and the restriction i2 + 1 = 0. Note that the number a + 0i, where a ∈ R, behaves like the real number a. Remark. What we have said in the last paragraph basically amounts to the following. Consider two complex numbers a + bi and c + di, where a, b, c, d ∈ R. We have the addition rule (a + bi) + (c + di) = (a + c) + (b + d)i, and the multiplication rule (a + bi)(c + di) = (ac − bd) + (ad + bc)i. A simple consequence is the subtraction rule (a + bi) − (c + di) = (a − c) + (b...
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