Particular Symbols β„• π•Ž β„€ β„š 𝕋 ℝ 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, …}
Integers. β„€={…, -2, -1, 0, 1, 2, 3, …}
Rational numbers. β„š is the set of fractions of integers. That is, the numbers contained in β„š are exactly those of the form n/m where n and m are integers and mβ‰ 0.

For example, β…“βˆˆβ„š and (-7)/12βˆˆβ„š.
Real numbers. ℝ 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€œthe number line”.

real number line

√2 and Ο€ are two of very many real numbers that are not rational numbers.
(Aside: the definition of ℝ above isn’t very precise, and thus isn’t a very good definition. The set of real numbers has a better definition, but it’s outside the scope of this course. For this semester We’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€œbarred” upper case letters, is used to denote these fundamental sets of numbers.

Definition 1
β€’ Ø denotes the empty set { }, which does not contain any elements.
β€’ β„• denotes the set of natural numbers { l, 2, 3, …}.
β€’ β„€ denotes the set of integers {…, -3, -2, -1, 0, 1, 2, 3, …}.
β€’ β„š denotes the set of rational numbers {p/q : p,qβˆˆβ„€ with qβ‰ 0}.
β€’ ℝ 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, bβˆˆβ„ with i=√(-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€œβ„•, ” the English words β€œthe natural numbers, ” and the set-theoretic notation β€œ{1, 2, 3, …}.” 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€œdecorated” with the superscripts β€œβˆ—β€ β€œ+, ” and β€œβ€”β€ to designate the corresponding subcollections of nonzero, positive, and negative numbers, respectively. For example, applying this symbolism to the integers β„€ = {…, -3, -2, -1, 0, 1, 2, 3, …}, we have

β„€* = {…, -3, -2, -1, 1, 2, 3, …},
β„€+ = {1, 2, 3, …},
β„€ = {-1, -2, -3, …}.

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€œcounting” numbers 1, 2, 3, . .. (as we have done here) and refer to the set {0, 1, 2, 3, …} as the set of whole numbers. On the other hand, many mathematicians think of zero as a β€œnatural” 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, …}.

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: Ø={}
β€’ The natural numbers: β„•={1,2,3,4,5,…}
β€’ The integers: β„€={…,-3,-2,-1,0,1,2,3,4,5,…}
β€’ The rational numbers: β„š={x∢x=m/n, where m,n∈ β„€ and nβ‰ 0}
β€’ The real numbers: ℝ (the set of all real numbers on the number line)
Notice that β„š is the set of all numbers that can be expressed as a fraction of two integers. You are surely aware that β„šβ‰ β„, as √2βˆ‰ β„š but √2βˆˆβ„.

🍫 Some examples of sets used particularly in Mathematics are

β„• : the set of all natural numbers. {1,2,3,…}
β„€ : the set of all integers. {…,-3,-2,-1,0,1,2,3,…}
β„š : the set of all rational numbers.
{…,-ΒΎ,…,-β…”,…,-Β½,…,Β½,…,β…”,…,ΒΎ,…}

ℝ : the set of real numbers.
(rational and irrational numbers)

β„€+ : the set of positive integers. {1,2,3,…}
β„š+ : the set of positive rational numbers.
{Β½,…,β…”,…,ΒΎ,…,β…˜,…}

ℝ+ : 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.
🍁 Numbers as Subsets of Real Number Set in a Venn Diagram

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