Number Systems

Mar 19, 2020

by Nandita Bhaskhar

tech-notes number-theory algebra math short-read

We start with our set of Natural numbers \(\mathbb{N} = {1,2,3,...}\) as the base. Gains represented by ✅ and losses by ⛔.

Property	\(\mathbb{N}\)	\(\mathbb{Z}\)	\(\mathbb{Q}\)	\(\mathbb{R}\)	\(\mathbb{C}\)
\(a + x = b\) is solvable	⛔	✅	✅	✅	✅
If \(a>b\), then \(xa>xb\) for all \(x\)	✅	⛔	⛔	⛔	⛔
\(ax = b\) is solvable	⛔	⛔	✅	✅	✅
There exists a next greater number for every element (induction)	✅	✅	⛔	⛔	⛔
There is no gap in the geometrical representation of the system	⛔	⛔	⛔	✅	✅
The system is countable	✅	✅	✅	⛔	⛔
All algebraic equations are solvable	⛔	⛔	⛔	⛔	✅
There is an ordering, i.e. can say \(a > b\)	✅	✅	✅	✅	⛔

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If you want to use parts of the text, any of the figures or share the article, please cite it as:

@article{ nanbhas2020numsys,
  title   = "Number Systems",
  author  = "Bhaskhar, Nandita",
  journal = "Blog: Roots of my Equation (web.stanford.edu/~nanbhas/blog/)",
  year    = "2020",
  url     = "https://web.stanford.edu/~nanbhas/blog/number-systems/"
}