6.001 Structure and Interpretation of Computer Programs. Copyright o 2004 by Massachusetts Institute of Technology 6.001 Notes: Section 9.1 Slide 9.1.1 Manipulating complex numbers In the last lecture, we introduced symbols into our language And we showed how to intermix them with numbers to create magnitude mixed expressions. We saw how to use that idea to create a mbolic differentiation program that could reason about algebraic expressions, rather than just numeric expressions. In this lecture, we are going to take the idea of symbols, and combine them with lots of different data structures to create tagged data structures Why do we need a tag?. and what is a tag? We'll answer these questions by considering an example. Suppose I want to build a 6001 SICP system to manipulate complex numbers. Remember that a complex number is a number with two parts, standardly epresented as a real and an imaginary part. Think of these as two axes in a vector space, or as a point in the plane (though we will see that there are special rules for how to manipulate such points that are different from normal vectors) Because we can represent a complex number as a vector, we can also think about representing such numbers in terms of a magnitude(or length) and an angle(typically with respect to the real axis) Manipulating complex numbers Slide 9.1.2 Now, let's assume we have some data abstractions for complex numbers. We have a constructor, and appropriate selectors, Real, imag, mag angle ncluding selectors for both the real and imaginary part, and for he magnitude and angle As we saw earlier, given such constructors and selectors, we can proceed to write code to manipulate complex numbers, without worrying about internal details. So let's do that C001 SICP6.001 Structure and Interpretation of Computer Programs. Copyright © 2004 by Massachusetts Institute of Technology. 6.001 Notes: Section 9.1 Slide 9.1.1 In the last lecture, we introduced symbols into our language. And we showed how to intermix them with numbers to create mixed expressions. We saw how to use that idea to create a symbolic differentiation program that could reason about algebraic expressions, rather than just numeric expressions. In this lecture, we are going to take the idea of symbols, and combine them with lots of different data structures, to create tagged data structures. Why do we need a tag? .. and what is a tag? We'll answer these questions by considering an example. Suppose I want to build a system to manipulate complex numbers. Remember that a complex number is a number with two parts, standardly represented as a real and an imaginary part. Think of these as two axes in a vector space, or as a point in the plane (though we will see that there are special rules for how to manipulate such points that are different from normal vectors). Because we can represent a complex number as a vector, we can also think about representing such numbers in terms of a magnitude (or length) and an angle (typically with respect to the real axis). Slide 9.1.2 Now, let's assume we have some data abstractions for complex numbers. We have a constructor, and appropriate selectors, including selectors for both the real and imaginary part, and for the magnitude and angle. As we saw earlier, given such constructors and selectors, we can proceed to write code to manipulate complex numbers, without worrying about internal details. So let's do that