# Particular Symbols β π β€ β π β β Represent Number Sets in Mathematics

Numbers

Among the most common sets appearing in math are sets of numbers. There are many different kinds of numbers. Below is a list of those that are most important for this course.
Natural numbers. β={1, 2, 3, 4, β¦}
Integers. β€={β¦, -2, -1, 0, 1, 2, 3, β¦}
Rational numbers. β is the set of fractions of integers. That is, the numbers contained in β are exactly those of the form n/m where n and m are integers and mβ 0.

For example, βββ and (-7)/12ββ.
Real numbers. β is the set of numbers that can be used to measure a distance, or the negative of a number used to measure a distance. The set of real numbers can be drawn as a line called βthe number lineβ.

β2 and Ο are two of very many real numbers that are not rational numbers.
(Aside: the definition of β above isnβt very precise, and thus isnβt a very good definition. The set of real numbers has a better definition, but itβs outside the scope of this course. For this semester Weβll make due with this intuitive notion of What a real number is.)

Certain sets are of widespread interest to mathematicians. Most likely, they are already familiar from your previous mathematics courses. The following notation, using βbarredβ upper case letters, is used to denote these fundamental sets of numbers.

Definition 1
β’ Γ denotes the empty set { }, which does not contain any elements.
β’ β denotes the set of natural numbers { l, 2, 3, β¦}.
β’ β€ denotes the set of integers {β¦, -3, -2, -1, 0, 1, 2, 3, β¦}.
β’ β denotes the set of rational numbers {p/q : p,qββ€ with qβ 0}.
β’ β denotes the set of real numbers consisting of directed distances from a designated point zero on the continuum of the real line.
β’ β denotes the set of complex numbers {a+bi : a, bββ with i=β(-1)}.

In this definition, various names are used for the same collection of numbers. For example, the natural numbers are referred to by the mathematical symbol ββ, β the English words βthe natural numbers, β and the set-theoretic notation β{1, 2, 3, β¦}.β Mathematicians move freely among these different ways of referring to the same number system as the situation warrants. In addition, the mathematical symbols for these sets are βdecoratedβ with the superscripts βββ β+, β and βββ to designate the corresponding subcollections of nonzero, positive, and negative numbers, respectively. For example, applying this symbolism to the integers β€ = {β¦, -3, -2, -1, 0, 1, 2, 3, β¦}, we have

β€* = {β¦, -3, -2, -1, 1, 2, 3, β¦},
β€+ = {1, 2, 3, β¦},
β€ = {-1, -2, -3, β¦}.

There is some discussion in the mathematics community concerning whether or not zero is a natural number. Many define the natural numbers in terms of the βcountingβ numbers 1, 2, 3, . .. (as we have done here) and refer to the set {0, 1, 2, 3, β¦} as the set of whole numbers. On the other hand, many mathematicians think of zero as a βnaturalβ number. For example, the axiomatic definition of the natural numbers introduced by the Italian mathematician Giuseppe Peano in the late 1800s includes zero. Throughout this book, we use Definition 1 and refer to the natural numbers as the set β = {1, 2, 3, β¦}.

We close this section with a summary of special sets. These are sets or types of sets that come up so often that they are given special names and symbols.
β’ The empty set: Γ={}
β’ The natural numbers: β={1,2,3,4,5,β¦}
β’ The integers: β€={β¦,-3,-2,-1,0,1,2,3,4,5,β¦}
β’ The rational numbers: β={xβΆx=m/n, where m,nβ β€ and nβ 0}
β’ The real numbers: β (the set of all real numbers on the number line)
Notice that β is the set of all numbers that can be expressed as a fraction of two integers. You are surely aware that ββ β, as β2β β but β2ββ.

π« Some examples of sets used particularly in Mathematics are

 β : the set of all natural numbers. {1,2,3,β¦} β€ : the set of all integers. {β¦,-3,-2,-1,0,1,2,3,β¦} β : the set of all rational numbers. {β¦,-ΒΎ,β¦,-β,β¦,-Β½,β¦,Β½,β¦,β,β¦,ΒΎ,β¦} β : the set of real numbers. (rational and irrational numbers) β€+ : the set of positive integers. {1,2,3,β¦} β+ : the set of positive rational numbers. {Β½,β¦,β,β¦,ΒΎ,β¦,β,β¦} β+ : the set of positive real numbers. (positive rational and irrational numbers)

The symbols for the special sets given above will be referred to throughout this text.
π Numbers as Subsets of Real Number Set in a Venn Diagram