Complex Numbers - Complex Numbers Advanced Level Pure...

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Unformatted text preview: Complex Numbers Advanced Level Pure Mathematics Advanced Level Pure Mathematics Algebra Chapter 10 Complex Numbers 10.1 Introduction 2 10.2 Geometrical Representation of a Complex Number 4 10.3 Polar Form of a Complex Number 4 10.4 Complex conjugate 8 10.5 Geometrical Applications 11 10.6 Transformation 19 10.7 DeMoivre's Theorem and nth Roots of a Complex Number 22 10.1 Introduction A Fundamental Concepts (1) A complex number z is a number of the form bi a + where a, b are real numbers and 1 2- = i . (2) The set C of all complex numbers is defined by Prepared by K. F. Ngai Page 1 1 2- = i 10 Complex Numbers Advanced Level Pure Mathematics C = { } 1 and , : 2- = ∈ + i R b a bi a where a is called the real part of z and ) Re( z a = and b is called the imaginary part of z and ) Im( z b = . (3) z is said to be purely imaginary if and only if ) Re( = z and ) Im( ≠ z . (4) When ) Im( = z , the complex number z is real . N.B. 1 , , 1 , 2 4 6 4 5 2 2 4 2 3- = ⋅ = = ⋅ = = ⋅ =- = ⋅ = i i i i i i i i i i i i i i . Example Solve 1 2 = + + x x in terms of i . Solution B Operations On Complex Numbers Let bi a z + = 1 and di c z + = 2 . Then (1) i d b c a z z ) ( ) ( 2 1 + + + = + (2) i d b c a z z ) ( ) ( 2 1- +- =- (3) 2 1 z z = ) )( ( di c bi a + + = i bc ad bd ac ) ( ) ( + +- (4) 2 1 z z = di c di c di c bi a-- ⋅ + + = i d c ad bc d c bd ac 2 2 2 2 ) ( +- + + + , where 2 ≠ z . N.B.(i) i i i i i- =- = = 1 1 2 ; (ii) 2 2 2 2 2 2 1 1 1 z d c d c di c di c z + = +- = + = . Example If i z 3 2 1 +- = and i z 4 1 2- = , find (a) 2 1 2 z z + (b) 1 2 iz z- (c) 2 1 z z (d) 2 1 z z Solution Example Express the following in the form of yi x + , where y x , are real numbers. (a) , 2 2 z z- + where i z + = 1 Prepared by K. F. Ngai Page 2 Complex Numbers Advanced Level Pure Mathematics (b) θ θ θ θ 2 sin 2 cos 1 2 sin 2 cos 1 i i + +- + (c) θ θ 2 sin 2 cos 1 1 i + + Solution Example Find the square roots of the complex number i 4 3 + and i 12 5- . Solution 10.2 Geometrical Representation of a Complex Number From the definition of complex numbers, a complex number bi a z + = is defined by the two real numbers a and b . Hence, if we consider the real part a as the coordinate x- in the rectangular coordinates system and the imaginary part b as the , coordinate y- then the complex number z can be represented by the point ) , ( b a on the plane. This plane is called the complex plane or the Argand diagram . On this plane, real numbers are represented by points on axis x- which is called the real axis ; imaginary numbers are Prepared by K. F. Ngai Page 3 Complex Numbers Advanced Level Pure Mathematics represented by points on the axis y- which is called the imaginary axis . The number 0 is represented by the origin O ....
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This note was uploaded on 11/26/2011 for the course COMPUTER S 1003 taught by Professor Angelosstavrou during the Spring '11 term at King Saud University.

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Complex Numbers - Complex Numbers Advanced Level Pure...

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