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How Do Different Categories of Numbers Compare To Each Other?

How do different categories of numbers compare to each other?

Summary

  1. Natural numbers are the counting numbers: 1, 2, 3, 4, ...
  2. Whole numbers are all the natural numbers including 0: 0, 1, 2, 3, ...
  3. Integers are the whole numbers plus negative counterparts: ..., -2, -1, 0, 1, 2, ....
  4. Rational numbers are numbers that you can write as fractions: 1/2, 1/3, -4/9, 1.75, 0.1111
  5. Irrational numbers are non-repeating, never-ending decimals, like π and √2
  6. Real numbers are all numbers found on a number line

Notes

    1. These are the numbers 1, 2, 3, 4, and so on
    1. So instead of starting at 1, we start at 0
    2. Whole numbers are 0, 1, 2, 3, and so on
    1. So we have 0, 1, 2, 3 and so on
    2. But we also have -1, -2, -3 and so on with negatives as well
    1. Rational numbers are numbers that can be written as fractions
    2. So any number you can rewrite as a/b, where b is not 0, is a rational number
    3. 1.75 can be rewritten as 7/4, so it is a rational number
    4. 0.1111... can be rewritten as 1/9, so it is also rational
    1. Irrational numbers are non-repeating, never-ending decimals
    2. They CANNOT be rewritten as fractions
    1. This will help us understand how these categories relate to each other
    1. Remember, these are the counting numbers: 1, 2, 3, and so on
    1. So these are the numbers 0, 1, 2, 3, and so on
    2. All natural numbers are also whole numbers
    3. So the box for natural numbers is completely contained in the box for whole numbers
    1. Integers are the numbers 0, 1, 2, 3, and so on, but also include -1, -2, -3 and so on with the negatives
    2. This means that all whole numbers and natural numbers are also integers
    1. Rational numbers are numbers that can be rewritten as fractions
    2. You can always rewrite an integer as itself over 1, so all integers are rational numbers
    3. This means all natural and whole numbers are rational numbers as well
    1. Numbers are either irrational or rational, but they cannot be both!
    2. Remember, these are never-ending, non-repeating decimals, like π and √2
    3. All of the numbers in the other categories so far were rational numbers, which means they are NOT irrational
    1. Real numbers include ALL rational and ALL irrational numbers
    1. So natural numbers are also whole numbers, integers, and rational numbers
    2. Whole numbers are also integers and rational numbers
    3. Integers are also rational numbers
    4. And all the numbers in the diagram are real numbers
    1. For example, not all whole numbers are natural numbers, since 0 is a whole number but not a natural number