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Unformatted text preview: 0 implies automatically that a ∈ R , not C . Earlier: C is complete, but not ordered. Theorem 7.1.2. For any real numbers a and b , ( a, 0) + ( b, 0) = ( a + b, 0) ( a, 0) · ( b, 0) = ( ab, 0) . This allows us to identify R with the subﬁeld of C consisting of the elements ( x, 0). C has another useful operation. Deﬁnition 7.1.3. z is the conjugate of z , deﬁned by z = x + y i = xy i . This corresponds to reﬂection in the horizontal axis R . Note: : C → C is a continuous function. Theorem 7.1.4. For z,w ∈ C , 1. z + w = z + w , 2. zw = z w , 3. if z = x + i y , then z + z = 2Re( z ) = 2 x and zz = 2 i Im( z ) = 2 y , 4. z z =  z  2 ≥ , with equality iﬀ z = 0 ....
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 Fall '11
 Wong
 Vector Space, Complex Numbers, Sequences And Series, Complex number, x1 + x2, vector space structure

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