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. ℕ={1, 2, 3, 4, …}
Integers. ℤ={…, -2, -1, 0, 1, 2, 3, …}
Rational numbers. ℚ is the set of fractions of integers. That is, the numbers contained in ℚ are exactly those of the form n/m where n and m are integers and m≠0.

For example, ⅓∈ℚ and (-7)/12∈ℚ.
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 “the number line”.

√2 and π 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 “barred” upper case letters, is used to denote these fundamental sets of numbers.

Definition 1
• Ø denotes the empty set { }, which does not contain any elements.
• ℕ denotes the set of natural numbers { l, 2, 3, …}.
• ℤ denotes the set of integers {…, -3, -2, -1, 0, 1, 2, 3, …}.
• ℚ denotes the set of rational numbers {p/q : p,q∈ℤ with q≠0}.
• ℝ denotes the set of real numbers consisting of directed distances from a designated point zero on the continuum of the real line.
• ℂ denotes the set of complex numbers {a+bi : a, b∈ℝ 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 “ℕ, ” the English words “the natural numbers, ” and the set-theoretic notation “{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 “decorated” with the superscripts “∗” “+, ” and “—” to designate the corresponding subcollections of nonzero, positive, and negative numbers, respectively. For example, applying this symbolism to the integers ℤ = {…, -3, -2, -1, 0, 1, 2, 3, …}, we have

ℤ^* = {…, -3, -2, -1, 1, 2, 3, …},
ℤ⁺ = {1, 2, 3, …},
ℤ^– = {-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 “counting” 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 “natural” 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 ℕ = {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.
• The empty set: Ø={}
• The natural numbers: ℕ={1,2,3,4,5,…}
• The integers: ℤ={…,-3,-2,-1,0,1,2,3,4,5,…}
• The rational numbers: ℚ={x∶x=m/n, where m,n∈ ℤ and n≠0}
• The real numbers: ℝ (the set of all real numbers on the number line)
Notice that ℚ is the set of all numbers that can be expressed as a fraction of two integers. You are surely aware that ℚ≠ℝ, as √2∉ ℚ but √2∈ℝ.

🍫 Some examples of sets used particularly in Mathematics are

ℕ : the set of all natural numbers. {1,2,3,…} ℤ : the set of all integers. {…,-3,-2,-1,0,1,2,3,…} ℚ : the set of all rational numbers. {…,-¾,…,-⅔,…,-½,…,½,…,⅔,…,¾,…} ℝ : the set of real numbers. (rational and irrational numbers) ℤ+ : the set of positive integers. {1,2,3,…} ℚ+ : the set of positive rational numbers. {½,…,⅔,…,¾,…,⅘,…} 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