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Miloslav Ciz 2024-08-31 14:44:45 +02:00
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real_number.md
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@ -30,7 +30,7 @@ Another **cool view of real numbers** is this: imagine fractions (rational numbe
```
...^ numerator
...^ numerator
6 |. . . . . . . __/ 2/3 = 4/6 = 8/12 = ...
5 |. . . . . .__/
4 |. . . . __/ .
@ -46,4 +46,4 @@ us -> 0 +-------------> denominator
From our point of view we can see all number, not just the fractions (which only sit on the integer grid points) -- all numbers, including real numbers, project to our field of view. Here fractions represent all the GRID points we see, i.e. a very dense set of points, however there are still gaps shining through which represent the real numbers that aren't fractions -- for example [pi](pi.md); if we shoot a ray from our standpoint in the exact angle that represents pi, the ray will go on forever without ever hitting any grid point! Such line will nearly miss some points, such as 355/113, which represents a good approximation of pi, but it will never hit any point exactly. So real numbers here are represented by the WHOLE, CONTINUOUS field of view.
**Are there bigger sets than those of real numbers?** Of course, a superset of real number is e.g. that of [complex numbers](complex_number.md) and [quaternions](quaternion.md), though they still have the same cardinality. But there are even sets that have bigger cardinality than reals, e.g. the set of all subsets of real numbers (so called [power set](power_set.md) of real numbers). In fact there are infinitely many such infinities of different cardinality.
**Are there bigger sets than those of real numbers?** Of course, a superset of real number is e.g. that of [complex numbers](complex_number.md) and [quaternions](quaternion.md), though they still have the same cardinality. But there are even sets that have bigger cardinality than reals, e.g. the set of all subsets of real numbers (so called [power set](power_set.md) of real numbers). In fact there are infinitely many such infinities of different cardinality.