Number Theory

The concept of “numbers” is itself very perplexing, and shows the evolution of mathematics from simple arithmetic,  to very complex operations.  Here is a brief overview of the types of numbers:

Each type of numbers in the list includes all the types listed above it.

  • N = Natural Numbers = {0, 1, 2, …}. These are also known as the “counting numbers”. Sometimes zero is excluded from this set, but included in the “Whole” numbers.
  • Z = Integers = {…, -2, -1, 0, 1, 2, …}. This set is the union of Natural Numbers and their counterparts, negative numbers.
  • Q = Rational Numbers = {i/j} for all ‘i’ and ‘j’ in the Integers, excluding any fraction with j=0. There is a lot to be said in the section below.
  • Irrational Numbers are all “Real” numbers that cannot be expressed as a ratio of integers. This is a poor definition, because it refers forward to the next set. The previous article explained why the square root of two is irrational, but did not explain the whole set. See the notes below, both for “square root” and especially for Irrational numbers. The above sets set are not included in this one.
  • R = Real Numbers consist of the union of all Rational and Irrational numbers.
  • Imaginary Numbers are required to solve problems involving the square root of negative numbers.
  • Infinite numbers express a cardinality for infinite sets of numbers. The above sets set are not included in this one.

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  1. nice


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