The set of real numbers consists of all the numbers on a number line. Subsets can include any collection of numbers, but the elements of an important subset should at least have several characteristics in common. Most of these subsets are only useful for specific calculations, but there are a few that have interesting properties and that help in understanding how the real number system works.
TL;DR (Too Long; Didn't Read)
The most important subsets of the set of real numbers include the rational and the irrational numbers. The set of rational numbers can be divided into further subsets, including the natural numbers, the whole numbers and the integers. Other subsets of the real numbers are the even and odd numbers, the prime numbers and the perfect numbers. Altogether there is an infinite number of subsets of the real numbers.
Real Number Subsets in General
For any set containing a quantity of n elements, the number of subsets is 2n. The set of real numbers has an infinite number of elements, and therefore the corresponding exponential of 2 is also infinite, giving an infinite numbers of subsets.
Many of these subsets can be used when working with the real number system and during calculations, but they are only useful for specific purposes. For example, for calculating the price of several pizzas for friends, only the subset of numbers from ten to one hundred may be of interest. An outdoor thermometer may only show the subset of temperatures from minus 40 to plus 120 degrees Fahrenheit. Working with subsets like these is useful because any result outside the expected subset is probably wrong.
The more general subsets of real numbers classify numbers according to their characteristics, and these subsets have unique properties as a result. The real number system evolved from subsets such as the natural numbers, which are used for counting, and such subsets form the basis for an understanding of algebra.
Subsets That Make Up the Real Numbers
The set of real numbers is made up of the rational and the irrational numbers. Rational numbers are integers and numbers that can be expressed as a fraction. All other real numbers are irrational, and they include numbers such as the square root of 2 and the number pi. Because irrational numbers are defined as a subset of real numbers, all irrational numbers must be real numbers.
Rational numbers can be divided into additional subsets. The natural numbers are numbers that were historically used in counting, and they are the sequence 1, 2, 3, etc. Whole numbers are the natural numbers plus zero. Integers are the whole numbers plus the negative natural numbers.
Other subsets of the rational numbers include such concepts as even, odd, prime and perfect numbers. Even numbers are integers that have 2 as a factor; odd numbers are all the other integers. Prime numbers are integers that have only themselves and 1 as factors. Perfect numbers are integers whose factors add up to the number. The smallest perfect number is 6 and its factors, 1, 2 and 3 add up to 6.
In general, calculations carried out with real numbers give real number answers, but there is an exception. There is no real number that, when multiplied by itself, gives a negative real number as an answer. As a result, the square root of a negative real number can't be a real number. The square roots of negative real numbers are called imaginary numbers, and they are the elements of a set of numbers completely separate from the real numbers.
The study of the subsets of real numbers is part of number theory, and it classifies numbers to make it easier to understand how number theory works. Becoming familiar with real number subsets and their properties is a good basis for further mathematical studies.