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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 subeld of C consisting of the elements ( x, 0). C has another useful operation. Denition 7.1.3. z is the conjugate of z , dened by z = x + y i = x-y i . This corresponds to reection 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 z-z = 2 i Im( z ) = 2 y , 4. z z = | z | 2 , with equality i z = 0 ....

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