# complex number - Complex numbers 1 Definition and notations...

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Complex numbers. 1. Definition and notations A complex number z is a formal expression of the form z = x + yi where x, y R . Denote by C = { x + yi | x, y R } the set of all complex numbers. For a complex number z = x + yi C , where x, y R put Re ( z ) := x, the real part of z, Im ( z ) := y, the imaginary part of z, | z | := p x 2 + y 2 , the absolute value of z, z := x - yi, the complex conjugate of z. By convention, for x R we identify z = x +0 i C with x R and write x + 0 i = x . Thus we have R C and, moreover, R = { z C | Im ( z ) = 0 } . Also by convention we write i = 0+1 i , - i = 0+( - 1) i , and, more generally, yi = 0 + yi where y R . Thus i, - i, 2 i, 5 i C . Note that for z C we have | z | = 0 if and only if z = 0. 2. Algebraic operations on complex numbers 2.1. Addition, multiplication and complex conjugation. For z 1 = x 1 + y 1 i, z 2 = x 2 + y 2 i C (where x 1 , x 2 , y 1 , y 2 R ) define z 1 + z 2 := ( x 1 + x 2 ) + i ( y 1 + y 2 ) z 1 z 2 := x 1 x 2 - y 1 y 2 + i ( x 1 y 2 + x 2 y 1 ) , so that z 1 + z 2 C and z 1 z 2 C . Note that, following the above definition of multiplication for complex numbers, we have i 2 = (0 + 1 i )(0 + 1 i ) = (0 2 - 1 2 ) + i (0 · 1 + 1 · 0) = - 1 + 0 i = - 1 , so that i 2 = - 1 in C . Also, for z = x + iy C (where x, y R ) put - z := - x + ( - y ) i, so that - z = ( - 1) z . 2.2. Basic properties of addition, multiplication and complex con- jugation. We have: 1
2 z 1 + ( z 2 + z 3 ) = ( z 1 + z 2 ) + z 3 for any z 1 , z 2 , z 3 C , z 1 + z 2 = z 2 + z 1 for any z 1 , z 2 C , z + 0 = 0 + z = z for any z C , z + ( - z ) = ( - z ) + z = 0 for any z C , z 1 ( z 2 z 3 ) = ( z 1 z 2 ) z 3 for any z 1 , z 2 , z 3 C , z 1 z 2 = z 2 z 1 for any z 1 , z 2 C , 1 · z = z · 1 = z for any z C , 0 · z = z · 0 = 0 for any z C , z 1 ( z 2 + z 3 ) = z 1 z 2 + z 1 z 3 for any z 1 , z 2 , z 3 C , z z = zz = x 2 + y 2 = | z | 2 R for any z = x + iy C , where x, y R , z z = 0 ⇐⇒ z = 0 for any z C , z = z for any z C , z 1 + z 2 = z 1 + z 2 for any z 1 , z 2 C , z 1 z 2 = z 1 z 2 for any z 1 , z 2 C , z = z ⇐⇒ z R where z C . 2.3. Division of complex numbers. Recall that for any z C we have z z = | z | 2 and that z z = | z 2 | >