What are Subsets of Real Numbers?

Real numbers are numbers that represent negative numbers, positive numbers, and zero. The set of real numbers comprises all the numbers on a number line. The subsets of real numbers may cover several sets of numbers, but each subset should have at least some common characteristics. Although these subsets are beneficial for performing some specific calculations, there are a few with some interesting properties that benefit in learning the functionality of a real number system.

The most fundamental subsets of the set of real numbers include the rational and the irrational numbers. The set of rational numbers can be further classified into different subsets that include natural numbers, whole numbers, and integers. There are some more subsets of the real numbers like even and odd numbers, prime numbers, and perfect numbers. Collectively there is an infinite number of subsets of the real numbers.

Rational Numbers

Rational Numbers can be defined as integers and numbers, which can be expressed as a fraction. Other real numbers are called irrational, and these comprise numbers such as the square root of 16 and the number pi. All irrational numbers are a subset of Real numbers; all irrational numbers are real numbers. Rational numbers can be further dissected into more subsets.

Natural Numbers

The natural numbers are part of the number system that was historically used and include all positive numbers starting from 1,2,3 etc. These numbers do not include zero; zero is considered as a part of the subset of whole numbers.

Whole Numbers

Whole numbers are a superset of natural numbers. Whole numbers include all the positive numbers starting from 1, 2, 3 upto infinity. The whole numbers set also includes zero. All the natural numbers are whole numbers. On the number system, whole numbers are plotted to start from 0,1,2 and so on. Some examples of the whole number are 0, 70, 98, 12345, etc.

Integers

Integers are numbers on both sides of zero, which means integers include the complete negative set of natural numbers and whole numbers that have zero and the positive set of natural numbers. Collectively integers are represented by the symbol Z in the number system. Few examples of Integers are -10, 5, 0, 23, 100 etc.

The rational numbers are further segregated based on different concepts such as prime numbers, odd numbers, even numbers, or perfect numbers.

Prime Numbers

What are Prime Numbers? – Prime numbers are integers that have only two factors, the number itself and number 1. The other sort of numbers are perfect numbers; these numbers have a unique property, the factors of perfect numbers always add up to the number itself. For example, the smallest perfect number is 6, and its factors are 1, 2, and 3, which add up to the number 6.

What are Even Numbers and Odd Numbers?

The concept behind even numbers is simple, any integer that has the number 2 as a factor is considered as an even number. Similarly, Odd numbers are all the other numbers that do not have two as a factor. Zero is the only integer that is neither Even nor Odd. In other terms, any whole number, which does not leave any remainder when divided by 2, is considered an Even number. A whole number is “even” if it is divisible by two without a remainder. For example, the number 10, when divided by the number 2, leaves no remainder and thus is an Even number. Few examples of Even numbers are 2,8,18,244 etc.

All the calculations based on real numbers will always result in a real number. But there are a few exceptions to this statement. No real number, when multiplied by itself, will ever generate a negative real number. Also, the square root of a negative number can never be a real number. The numbers created by rooting a negative real number are called an imaginary number, a whole new set of numbers.