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The Road to Reality Study Notes: Difference between revisions
→5.1 Geometry of complex algebra
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Penrose asks us to view complex addition and multiplication as transformations from the complex plane to itself, rather than just as simple addition and multiplication. The visual representations of these operations are given as the parallelogram and similar-triangle laws for addition and multiplication respectively. Rather than just ‘adding’ and ‘multiplying’ these can be viewed as ‘translation’ and ‘rotation’ within the complex plane. | Penrose asks us to view complex addition and multiplication as transformations from the complex plane to itself, rather than just as simple addition and multiplication. The visual representations of these operations are given as the parallelogram and similar-triangle laws for addition and multiplication respectively. Rather than just ‘adding’ and ‘multiplying’ these can be viewed as ‘translation’ and ‘rotation’ within the complex plane. | ||
[[File:Fig 5p1.png|thumb|center]] | [[File:Fig 5p1.png|thumb|center]] | ||
For example, multiply a real number by the complex number i rotates the point in the complex plane π/2. | For example, multiply a real number by the complex number $$i$$ rotates the point in the complex plane π/2. | ||
[[File:Fig 5p2.png|thumb|center]] | [[File:Fig 5p2.png|thumb|center]] | ||
Penrose further introduces the concept of polar coordinates where r is the distance from the origin and θ is the angle made from the real axis in an anticlockwise direction. | Penrose further introduces the concept of polar coordinates where $$r$$ is the distance from the origin and $$θ$$ is the angle made from the real axis in an anticlockwise direction. | ||
[[File:Fig 5p4.png|thumb|center]] | [[File:Fig 5p4.png|thumb|center]] | ||