# Squaresolid its absolute value is z radicalbig x 2 y

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squaresolid Its Absolute Value is | z | = radicalbig x 2 + y 2 . O z = x + iy x y | z | = radicalbig x 2 + y 2 square The Absolute Value of the number z is the length of the vector z , and it represents the distance from the origin O to the point z . 32 / 57 16
Argument squaresolid Let z = x + iy , where x, y R . | z | = radicalbig x 2 + y 2 . z = x + iy x y O | z | = radicalbig x 2 + y 2 θ square The Argument of z ( negationslash = 0) is the angle From the positive direction of the real axis To the vector z , denoted by θ = arg z . squaresolid cos θ = x | z | and sin θ = y | z | . squaresolid z = | z | parenleftbigg x | z | + i y | z | parenrightbigg = | z | (cos θ + i sin θ ) . square This is called the Polar Form of z . 33 / 57 Examples squaresolid Find the argument and the polar form of z = 1 + i . 1 1 O x y θ square We can see that arg z = θ = π 4 . But how to compute? squaresolid | z | = 1 2 + 1 2 = 2 . squaresolid cos θ = x | z | = 1 2 θ = cos - 1 parenleftbigg 1 2 parenrightbigg = π 4 . squaresolid Polar form: 1 + i = 2 parenleftBig cos π 4 + i sin π 4 parenrightBig . 34 / 57 17
Examples squaresolid Find the argument and the polar form of z = 1 i . - 1 1 O x y θ square | z | = radicalbig 1 2 + ( 1) 2 = 2 . squaresolid cos θ = x | z | = 1 2 . squaresolid y = 1 < 0 arg z = cos - 1 parenleftbigg 1 2 parenrightbigg = π 4 . square Polar form: 1 i = 2 bracketleftBig cos parenleftBig π 4 parenrightBig + i sin parenleftBig π 4 parenrightBigbracketrightBig . 35 / 57 Find Argument squaresolid Let z = x + iy negationslash = 0 , where x, y R . x = Re z and y = Im z . i) If Im z 0 , then arg z = cos - 1 (Re z/ | z | ) ; ii) If Im z < 0 , then arg z = cos - 1 (Re z/ | z | ) . squaresolid Examples . square Let z = 10 . Re z = 10 , Im z = 0 ; | z | = 10 . squaresolid arg z = cos - 1 (10 / 10) = cos - 1 (1) = 0 . squaresolid 10 = 10(cos 0 + i sin 0) . square Let z = 2 . Re z = 2 , Im z = 0 ; | z | = 2 . squaresolid arg z = cos - 1 ( 2 / 2) = cos - 1 ( 1) = π . squaresolid 2 = 2(cos π + i sin π ) . square Let z = 7 i . Re z = 0 , Im z = 7 ; | z | = 7 . squaresolid arg z = cos - 1 (0 / 7) = cos - 1 (0) = π/ 2 . squaresolid 7 i = 7[cos( π/ 2) + i sin( π/ 2)] . 36 / 57 18
Find Argument squaresolid Let z = x + iy negationslash = 0 , where x, y R . x = Re z and y = Im z . i) If Im z 0 , then arg z = cos - 1 (Re z/ | z | ) ; ii) If Im z < 0 , then arg z = cos - 1 (Re z/ | z | ) . squaresolid Examples . square Let z = 8 i . Re z = 0 , Im z = 8 ; | z | = 8 . squaresolid arg z = cos - 1 (0 / 8) = cos - 1 (0) = π/ 2 . squaresolid 8 i = 8[cos( π/ 2) + i sin( π/ 2)] . square Let z = 1 + i . Re z = 1 , Im z = 1 ; | z | = 2 . squaresolid arg z = cos - 1 ( 1 / 2) = 3 π/ 4 . squaresolid 1 + i = 2 [cos(3 π/ 4) + i sin(3 π/ 4)] . square Let z = 1 i . Re z = 1 , Im z = 1 ; | z | = 2 . squaresolid arg z = cos - 1 ( 1 / 2) = 3 π/ 4 . squaresolid 1 i = 2 [cos( 3 π/ 4) + i sin( 3 π/ 4)] . 37 / 57 Find Argument squaresolid Let z = x + iy negationslash = 0 , where x, y R . x = Re z and y = Im z . i) If Im z 0 , then arg z = cos - 1 (Re z/ | z | ) ; ii) If Im z < 0 , then arg z = cos - 1 (Re z/ | z | ) . squaresolid Properties . square If z is real and positive, arg z = 0 . square If z is real and negative, arg z = π . square If z is not real, (i.e., Im z negationslash = 0 ), squaresolid arg( z * ) = arg z . squaresolid Remark . The Argument is not unique. square If θ is an argument of z , then squaresolid θ +2 π , θ +4 π, . . . ; θ 2 π, θ 4 π, . . . are also arguments of z . square We usually take π < arg z π . 38 / 57 19
Multiplication squaresolid Let z 1 = x 1 + iy 1 and z 2 = x 2 + iy 2 , x 1 , x 2 , y 1 , y 2 R . square By noting that i 2 = 1 , we can find z 1 z 2 .