Types of numbers
Also known as: number classifications, number sets
The types of numbers are nested classifications of the real numbers: natural numbers, whole numbers, integers, rational numbers, and irrational numbers, with the rationals and irrationals together making up the reals.
Mathematicians sort numbers into named sets, each one containing the last. Natural numbers are the counting numbers: 1, 2, 3, and so on. Adding zero gives the whole numbers. Adding negatives gives the integers: ..., −2, −1, 0, 1, 2, .... Each set is a subset of the next, so every natural number is also a whole number and an integer.
The next expansion brings in fractions. A rational number is any number expressible as a fraction of two integers (with a nonzero denominator) — this includes all integers (5 = 5/1), fractions like 3/4, and any decimal that terminates or repeats, such as 0.25 or 0.333.... An irrational number cannot be written as such a fraction; its decimal form goes on forever without repeating. Famous examples are π and √2. Together, the rationals and irrationals make up the real numbers.
Other useful labels cut across these sets. Even integers are divisible by 2; odd integers are not. A prime number is a natural number greater than 1 whose only factors are 1 and itself (2, 3, 5, 7, 11, ...), while composite numbers have additional factors. Note that 2 is the only even prime, and 1 is neither prime nor composite.
Number classification questions appear on the ACT and CLT math sections, which ask you to identify which set a number belongs to, spot the exception in a list, or apply properties of evens, odds, and primes. Knowing the definitions cold turns these into quick points.
Key takeaways
- Natural numbers (1, 2, 3, ...) plus zero form the whole numbers; adding negatives forms the integers.
- Rational numbers can be written as a fraction of integers; their decimals terminate or repeat.
- Irrational numbers like π and √2 have non-repeating, non-terminating decimals.
- Rationals and irrationals together form the real numbers.
- 1 is neither prime nor composite, and 2 is the only even prime.