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Basic Mathematics - Serge Lang

Numbers

Natural Numbers

  • Before we go to Integers we have to discuss natural numbers
  • $\mathbb{N}$ is defined as numbers starting from 0, 1, 2, 3, …, to infinity
  • Yes, from 0, contrary to what you’ve been taught your whole life
  • Though there was confusion amongst mathematicians on whether 0 should be included or not, now-a-days $\mathbb{N}$ is canonically defined to include 0 ISO 80000-2
    • The reason for this confusion is that starting with 1 offers a nice property of constructing all other elements by just repeated addition
    • This will make more sense once you get into sets and groups and things of that nature
    • Having 0 messes this up. You no longer can have a singleton generator.
  • 0 is defined as the origin
    • You’ll see that having an origin is pretty important for many properties
    • So 0 being included will make a ton of sense
    • The class of number without 0 won’t be able to inherit any of these useful properties and hence won’t fit in the subset diagram that oh-so-common in Mathematics.

Integers

  • If we add (no pun intended) negative numbers to the collection of natural numbers, we get Integers.
  • We denote them using $\mathbb{Z}$ (why Z instead of I? The french apparently. $\mathbb{Q}$ and $\mathbb{Z}$ stands for Quotient and Zahlen in German. That’s where rational numbers too get Q from)