Number sets group numbers by shared properties. Natural numbers describe counting, integers add negative whole numbers, and rational numbers include ratios of integers. Irrational real numbers cannot be written as such ratios. Rational and irrational numbers together form the real numbers considered here.
How the number sets fit together
Every natural number is an integer, every integer is rational, and every rational number is real. Irrational numbers form the part of the real numbers outside the rationals. They do not sit inside the rational set.
Natural numbers, zero and integer subsets
This article includes zero among the natural numbers. Some courses start them at one, so state the convention rather than guessing. Integers extend in both directions without a first or last element. Even numbers, odd numbers and primes are subsets of the appropriate integer or natural set; 1 is not prime.
Symbols for “positive” subsets vary between books, especially concerning zero. Use a condition such as greater than zero or greater than or equal to zero when that distinction matters. Removing the singleton containing zero creates the nonzero subset.
Rational or irrational: check the value, not its appearance
A rational number is a ratio of integers with nonzero denominator. Its decimal expansion terminates or eventually repeats. An irrational decimal neither terminates nor eventually repeats. A few displayed digits followed by dots are not enough to prove irrationality.
Thus −3 is integer, rational and real. The square root of 9 is natural because its value is 3, whereas the square root of 2 is irrational. A radical sign alone does not determine the number’s class.
Open and closed endpoints
Square brackets include a finite endpoint; parentheses exclude it. The inequalities below make each convention explicit. An infinite endpoint is always open because infinity is not a real number included in an interval.
Worked example: integers from −2 up to, but not including, 3
The interval contains every real number satisfying its bounds. Intersecting it with the integers keeps only the five listed values. The interval itself is not merely that finite list.
Review set operations for intersection and inclusion, and divisibility rules for integer structure.
