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i 1 1 i i = i = = = i (i 2 ) ( 1) i i EXAMPLE 15 Find z if z( 7 z + 14 5i ) = 0 SOLUTION One obvious solution to the equation is z = 0 The other one is found to be 7 z + 14 5i = 0 7 z = 14 + 5i or 5 z = 2 + i 7
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Expansions of complex numbers can be written down immediately using Pascal s triangle, which lists the coef cients in an expansion of the form ( x + y )n We list the rst ve rows here: 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 (113)
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The rst row corresponds to ( x + y )0, the second row to ( x + y )1, and so on For example, looking at the third row we have coef cients 1, 2, 1 This means that ( x + y )2 = x 2 + 2 xy + y 2 EXAMPLE 16 Write ( 2 i )4 in the standard form a + ib
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SOLUTION The coef cients for the fourth power are found in row ve of Pascal s triangle In general: ( x + y )4 = x 4 + 4 x 3 y + 6 x 2 y 2 + 4 xy3 + y 4 Hence ( 2 i )4 = 2 4 + 4( 2 3 )( i ) + 6( 2 2 )( i )2 + 4( 2 )( i )3 + ( i )4 = 16 32i + 24( i )2 + 8( i )3 + ( i )4
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Now let s look at some of the terms involving powers of i individually First we have ( i )2 = ( 1)2 i 2 = ( +1)( 1) = 1 The last two terms are ( i )3 = ( 1)3 i 3 = ( 1)(i i 2 ) = ( 1)(i )( 1) = +i ( i )4 = [( i )2 ]2 = ( 1)2 = +1 Therefore we have ( 2 i )4 = 16 32i + 24( i )2 + 8( i )3 + ( i )4 = 16 32i 24 + 8i + 1 = 7 24i
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We have already seen some of the basics of how to handle complex numbers, like how to add or multiply them Now we state the formal axioms of the complex number system which allow mathematicians to describe complex numbers as a
Complex Numbers
eld These axioms should be familiar since their general statement is similar to that used for the reals We suppose that u, w, z are three complex numbers, that is, u, w, z Then these axioms follow: z+w and zw (closure law) (commutative law of addition) (114) (115) (116) (117) (118) (119)
z+w= w+z
u + ( w + z ) = (u + w) + z (associative law of additio) zw = wz u( wz ) = (uw) z u( w + z ) = uw + uz (commutative law of multiplication) c (associative law of multiplication) (distributive law)
The identity with respect to addition is given by z = 0 + 0i , which satis es z+0= 0+z (120)
The identity with respect to multiplication is given by z = 1 + i 0 = 1, which satis es z 1 = 1 z = z (121)
For any complex number z there exists an additive inverse, which we denote by z that satis es z + ( z) = ( z) + z = 0 (122)