Natural number

In mathematics, the natural numbers are the numbers used for counting up, starting from 0 or 1. The first few natural numbers are 0 (if included), 1, 2, 3, and so on. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set of the natural numbers is commonly denoted by a bold N or a blackboard bold ⁠ N {\displaystyle \mathbb {N} } ⁠. The natural numbers are used for counting, and for labeling the result of a count, such as: "there are seven days in a week", in which case they are called cardinal numbers. They are also used to label places in an ordered series, such as: "the third day of the month", in which case they are called ordinal numbers. Natural numbers are commonly expressed in writing using ten symbols called digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). These expressions, called numerals, can also be used as unique identifiers or labels (like the jersey numbers of a sports team) that are referred to as nominal numbers, which resemble natural numbers but have no specific mathematical properties. Natural numbers can be compared by magnitude, with larger numbers coming after smaller ones in the list 1, 2, 3, .... This gives a total order on the natural numbers. Two basic arithmetic operations are defined on natural numbers: addition and multiplication. However, the inverse operations, subtraction and division, only sometimes give natural-number results: subtracting a larger natural number from a smaller one results in a negative number and dividing one natural number by another commonly leaves a remainder. The most common number systems used throughout mathematics – the integers, rational numbers, real numbers, and complex numbers – contain the natural numbers, and are defined successively, one from the other, starting...