Number

A number is an abstract mathematical object hard to define. In mathematics, a number is formally a member of a given set (possibly an ordered set). It conveys the ideas of counting, ordering, and measurement. However, due to the expressiveness of positional number systems, the usefulness of geometric objects, and the advances in different scientific fields, it can convey more properties and can be expressed in different notations.

Numbers are used to count (e.g., there are 26 simple latin letters). Numbers can be compared (e.g., e is lower than pi in the real number set). In many natural sciences, they are used to measure (e.g., the weight of 50 lbs in imperial system is approximately equal to the mass of 22.7 kg in the metric system).

A word written only with digits is called a numeral, and may represent a number. Numerals are often used for labeling (like telephone numbers), for ordering (like serial numbers), and for encoding (like ISBNs).

The writing of a number depends on the numeral system in use. For instance, the number 12 is written "1100" in base 2, "C" in base 16, and "XII" as a roman numeral. We can geometrically represent a number with vectors in a cartesian system or by drawing simple shapes (e.g., squares and circles). There are other means to express a number.

Number sets

This section presents different number sets, but this list is not exhaustive.

1. The natural numbers ($\scriptstyle \mathbb {N}$) are used to count things (e.g., there are 52 weeks in a Julian year). This set contains many remarquable subsets : prime numbers, Fibonacci numbers, perfect numbers, catalan numbers, etc.
2. The integers ($\scriptstyle \mathbb {Z}$) express presence and lack of something, debits and credits, etc. (e.g., a company owes 60 millions US dollars to a bank). This set includes the natural numbers.
3. The rational numbers ($\scriptstyle \mathbb {Q}$) define a part of something (e.g., someone received half of its pay yesterday). This set includes the integers.
4. The irrational numbers ($\scriptstyle \mathbb {J}$) find application in many abstract mathematical fields, such as algebra and number theory. This set do not share any member with the rational number set.
5. The real numbers ($\scriptstyle \mathbb {R}$) find applications in measurements and advanced mathematics. They are usually best written as decimal numbers (e.g., the value of e is approximately equal to 2.718281828). This set includes the rational numbers and the irrational numbers.
6. The complex numbers ($\scriptstyle \mathbb {C}$) were discovered while searching solutions to some polynomials (e.g., the polynomial $\scriptstyle x^{2}+1=0$ has two solutions, one being $\scriptstyle {\sqrt {-1}}=(0,1)=i$). Because the complex number set is algebraically closed, it finds applications in many scientific fields, such as engineering and applied mathematics. This set includes the real numbers.
7. A complex number that is solution to a polynomial in integer coefficients is an algebraic number. This set includes all rational numbers and a subset of the irrational numbers. Any other complex number is a transcendental number.

In order to meet their needs, scientists created other number sets. To ease the study of quadratic forms, Carl Friedrich Gauss introduced from 1829 to 1831 what is known today as the gaussian integers. While studying 3D mechanics, William Rowan Hamilton introduced the quaternions in 1843 (today, they are largely superseded by vectors). Octonions were discovered in 1843. Georg Cantor, through its naive set theory, formally defined the notion of infinity in 1895. Kurt Hensel first described the p-adic numbers in 1897, looking for a way to bring the ideas and the techniques of power series within number theory.

We can consider vectors and matrices as number sets, since they mathematically abstract phenomenas in a unique way and we can apply operations upon them.

Notation

The notation plays a central role in the perception of what a number is and what we can do with it. A good notation saves lots of work when operating on numbers (in fact on any mathematical abstract objects). For instance, it is possible to add numbers written in roman numerals (e.g., MCMXCVIII plus CCXVII). However, it is faster to add numbers written in base 10 (e.g., 1998 plus 217). The gain is higher when multiplying numbers.