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Fall 2008Programming Development Techniques 1 Topic 12 Multiple Representations of Abstract Data – Complex Numbers Section 2.4.1

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Fall 2008Programming Development Techniques 2 Multiple representations for abstract data Implementation of complex numbers as an example Illustrates how one representation can be better for one operation, but another representation might be better for another operation (Scheme already has complex numbers, but we'll pretend that it doesn't)

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Fall 2008Programming Development Techniques 3 Complex numbers (math view) z = x + i y (rectangular form) = r e iα (polar form) x: real part of z y: imaginary part of z r: magnitude of z a: angle of z x Real y Imaginary r a Z

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Fall 2008Programming Development Techniques 4 Complex number arithmetic Addition – addition of coordinates – add real parts and imaginary parts z 1 + z 2 = x 1 + iy 1 + x 2 + iy 2 = (x 1 + x 2 ) + i(y 1 +y 2 ) Multiplication – easier to think of in polar form z 1 * z 2 = r 1 e iα ¹ * r 2 e iα ² =(r 1 * r 2 )e i(α ¹ + α ² )

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Fall 2008Programming Development Techniques 5 SO? There are two different representations of complex numbers. Some operations on complex numbers are easier to think of in terms of one operation and others in terms of the other representation. Yet all operations for manipulating complex numbers should be available no matter which representation is chosen. Want to have access to each part: real, imaginary, magnitude, angle no matter which representation is chosen.

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Fall 2008Programming Development Techniques 6 Two representations Rectangular make-from-real-imag - constructor real-part – selector imag-part – selector Polar make-from-mag-ang – constructor magnitude – selector angle – selector Two different representations possible for the same number.

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Fall 2008Programming Development Techniques 7 Addition ; adds together two complex numbers ; uses the representation of addition of coordinates ; in terms of real and imaginary parts (define (add-complex z1 z2) (make-from-real-imag (+ (real-part z1) (real-part z2)) (+ (imag-part z1) (imag-part z2))))

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Fall 2008Programming Development Techniques 8 Subtraction ; subtract one complex number from another ; uses the representation of subtraction of ; coordinates in terms of real and ; imaginary parts (define (sub-complex z1 z2) (make-from-real-imag (- (real-part z1) (real-part z2)) (- (imag-part z1) (imag-part z2))))

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Fall 2008Programming Development Techniques 9 Multiplication ; multiplies two complex numbers ; uses the representation as polar form ; in terms of magnitude and angle (define (mul-complex z1 z2) (make-from-mag-ang (* (magnitude z1) (magnitude z2)) (+ (angle z1) (angle z2))))

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Fall 2008Programming Development Techniques 10 Division ; divides one complex number from another ; uses the representation as polar form ; in terms of magnitude and angle (define (div-complex z1 z2) (make-from-mag-ang (/ (magnitude z1) (magnitude z2)) (- (angle z1) (angle z2))))

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Fall 2008Programming Development Techniques 11 Choose a representation We must implement constructors and selectors in terms of primitive numbers and primitive list structure. Which representation should we use?? Rectangular form (real part, imaginary part – good for addition and subtraction) Polar form (magnitude and angle – good for multiplication and division) Either representation OK as long as we can select out all of the pieces we need – real, imaginary, magnitude, angle

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Fall 2008Programming Development Techniques 12 Rectangular Representation ;; lower level implementation ; RECTANGULAR FORM REPRESENTATION ; takes a real and imaginary part and ; creates a complex number represented ; in rectangular form (define (make-from-real-imag x y) (cons x y))

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Fall 2008Programming Development Techniques 13 Rectangular Representation (cont) ; given an imaginary number in ; rectangular form ; returns the real part (define (real-part z) (car z)) ; given an imaginary number in ; rectangular form ; returns the imaginary part (define (imag-part z) (cdr z))

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Fall 2008Programming Development Techniques 14 Rectangular Representation (cont) ; given an imaginary number in rectangular form ; return the magnitude (using trigonomic rels) (define (magnitude z) (sqrt (+ (square (real-part z)) (square (imag-part z))))) ; given an imaginary number in rectangular form ; return the angle (using trigonomic rels) (define (angle z) (atan (imag-part z) (real-part z)))

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Fall 2008Programming Development Techniques 15 Rectangular Representation (cont) ; takes a magnigude and an angle and ; creates a complex number ; represented in rectangular form (define (make-from-mag-ang r a) (make-from-real-mag (* r (cos a)) (* r (sin a))))

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Fall 2008Programming Development Techniques 16 Polar representation ;; lower level implementation ; POLAR FORM REPRESENTATION ; takes a magnigude and an angle and ; creates a complex number represented ; in polar form (define (make-from-mag-ang r a) (cons r a))

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Fall 2008Programming Development Techniques 17 Polar Representation (cont) ; given an imaginary number in ; polar form ; return the magnitude (define (magnitude z) (car z)) ; given an imaginary number in ; rectangular form ; return the angle (define (angle z) (cdr z))

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Fall 2008Programming Development Techniques 18 Polar Representation (cont) ; given an imaginary number in ; polar form ; returns the real part ; (using trignomic rels) (define (real-part z) (* (magnitude z) (cos (angle z)))) ; given an imaginary number in ; polar form ; returns the imaginary part ; (using trigonomic rels) (define (imag-part z) (* (magnitude z) (sin (angle z))))

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Fall 2008Programming Development Techniques 19 Polar Representation (cont) ; takes a real and imaginary part and ; creates a complex number represented ; in polar form (harder) (define (make-from-real-imag x y) (make-from-mag-ang (sqrt (+ (square x) (square y))) (atan y x)))

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Fall 2008Programming Development Techniques 20 Which Representation? Note – either representation will work fine. Notice that some of the selectors/constructors are easier with one representation over the other But, no matter which is used, our basic operations will still work.

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Key Points Two complex numbers can be multiplied by expressing each number in the form z = x + jy, then using distributivity and the rule j2 = -1 (i.e.,

Key Points Two complex numbers can be multiplied by expressing each number in the form z = x + jy, then using distributivity and the rule j2 = -1 (i.e.,

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