(January 18, 2022 at 5:45 pm)brewer Wrote: If zero is not 'natural', where does than leave negative numbers?
Yeah Poly, I'm looking to you to hurt my math brain again.
The usual progression is something like this:
1. Natural numbers: 0,1,2,3,4,....
2. Integers: Either a natural number or the negative of a natural number, so ..,-3, -2, -1, 0, 1, 2, 3,...
3. Rational numbers: Fractions, one integer divided by a non-zero integer. 2/3, -5/8, 13/2, 4=4/1, -123/43,....
4. Real numbers: intuitively, numbers with (possibly infinite) decimal expansions. pi, e, sqrt(2), ....
5. Complex numbers. If i is the square root of -1, then a complex number is one of the form a+bi where a and b are real numbers.
These are the basic number systems, but there are a host of others.
Algebraic numbers: those that are roots of some polynomial with integer coefficients. sqrt(2) is algebraic. so is sqrt(2)+cbrt(5).
Gaussian integers: those complex numbers a+bi where a and b are integers. these have many properties in common with the integers.
These are the usual algebraic number systems.
But we can go in a different direction. All natural numbers are also 'cardinal numbers': these count the 'number of things' in sets. The natural numbers are, specifically, the
*finite* cardinal numbers.
But there are infinite cardinal numbers as well. These are studied in set theory and usually not as algebraic structures because they lack many of the 'nice' properties of the other systems above. if you are trying to say infinity is a number, this is what you are probably talking about.
Ordinal numbers: instead of the 'size' of a set, we put an 'order structure' on the set and compare different order structures. Usually, we do this with order structures that are 'nice' in the sense of being 'well-ordered' (a technical term that I can go into if anyone wants). The natural numbers are, again, the finite ordinal numbers. But ordinal numbers have a much more complex structure than cardinal numbers do. They also fail to obey certain fairly natural algebraic properties, so are usually studied by set theorists as well.
In standard set theory, following Von Neumann, every cardinal number is an ordinal number, but not vice versa.
