1.3 Sets of Numbers

In mathematics, we need to refer to some certain sets of numbers so often that we denote them by special symbols (in particular by $\mathbb{N},\mathbb{Z},\mathbb{Q},\mathbb{R}$, and $\mathbb{C}$). These double-struck letters (sometimes called blackboard bold letters) are used to distinguish these specific sets (defined below) from some other sets that happen to be denoted by the same letters, for example by $R$.

 

Natural Numbers

The most common numbers are the numbers 1, 2, 3, … which are used for counting are called natural numbers or positive integers. The set of all natural numbers is often denoted by $\mathbb{N}$

$$\mathbb{N}=\{1,2,3,\dots \}$$

Integers

The numbers $-1,-2,-3,\dots$ are called negative integers. The set of all integers (positive and negative and zero) is denoted by $\mathbb{Z}$ (standing for the German word Zahlen that means “numbers”):

$$\mathbb{Z}=\{\dots ,-3,-2,-1,0,1,2,3,\dots \}$$

which can also be written as

$$\mathbb{Z}=\{0,\pm 1,\pm 2,\pm 3,\pm 4,\dots \}.$$

Rational Numbers

A rational number is a number that can be written as a fraction, or quotient, of two integers. For example, $-3/4$ is a rational number.

• All integers are rational numbers because they can be written as a fraction with denominator 1; for example, $-3$ can be written as $-3/1$.
• Other examples of rational numbers include numbers that have decimal representations that either terminate (for example $3.89$ can be written as $389/100$) or do not terminate but have repeating blocks of digits (for example $0.3333\dots$ is the same as $1/3$).

The set of all rational numbers is denoted by $\mathbb{Q}$

$$\mathbb{Q}=\{\frac{p}{q}|p,q\in \mathbb{Z}\}.$$

Irrational Numbers

The ancient Greeks knew that the lengths of some lines in simple figures cannot be expressed as the ratio of integers. For example, from the Pythagorean theorem they knew that the diagonal of a square with sides of unit length is $\sqrt{2}$, but $\sqrt{2}$ cannot be written in the form of $p/q$ where $p$ and $q$ are integers. Another well-known example that cannot be expressed as the ratio of two integers is $\pi =3.141592\dots$. Such numbers are called irrational numbers.

• The decimal digit representation of an irrational number goes on forever and never repeats.

 

  Real Numbers

The set of all rational and irrational numbers is called the set of real numbers and is denoted by $\mathbb{R}$.

• Note that $$\mathbb{N}\subset \mathbb{Z}\subset \mathbb{Q}\subset \mathbb{R}.$$

 

Complex Numbers

The set of all numbers in the form $a+ib$ where $a$ and $b$ are real numbers and $i=\sqrt{-1}$ is called the set of complex numbers and is denoted by $\mathbb{C}$$$\mathbb{C}=\{a+ib|a,b=\mathbb{R}\}.$$

In this course, we only deal with real numbers.

 

Number Line 

Real numbers can be thought of as points on an infinitely long line called the number line (also known as number axis, coordinate line, or real line), where $0$ is the origin (see the following figure). The positive direction is indicated by an arrow. A unit length is assumed on this line and integers are multiples of this unit. Negative numbers occur to the left of zero. The number $0$ is neither positive nor negative.

 

 

Properties of Real Numbers

Here are some properties of real numbers:

• The product of any number and $0$ is $0$ $$0\cdot a=0\text{for all }a$$

 

• If $ab=0$ then either $a=0$ or $b=0$ (or both of them are zero).

 

• If $a\ne 0$ is a number then there is a unique number $b$ such that $ab=ba=1$, and we write$$b=\frac{1}{a}\text{or}b=a^{-1}.$$  We say that $b$ is the inverse of $a$. If $a$ and $b$ are two numbers $a/b$ means $a\cdot b^{-1}$.

 

• Note that $0$ has no inverse because $1/0=b$ must mean $1=0\cdot b$, but $0\cdot b=0\ne 1$. This shows that we cannot assign a definite value to $1/0$. So division by zero is not an admissible operation, and if $a$ is any number, $a/0$ has no meaning.

 

• If $a\ne 0$, then ${\displaystyle \frac{0}{a}}=0$ because $a\cdot 0=0$ (but again $a/0$ has no meaning).

 

• Basic properties of the fundamental operations — addition, subtraction, multiplication and division — that are often called the basic rules of algebra are summarized in the following table. These properties are true for real numbers, variables, and algebraic expressions.

 

Summary of Basic Rules of Algebra

 

Name	Math	Description	Example
Commutative Property of Addition	$a+b=b+a$	We can add numbers in any order	$2+3=3+2$ $4+x=x+4$
Commutative Property of Multiplication	$ab=ba$	We can multiply in any order	$2\cdot 3=3\cdot 2$ $x\cdot 4=4\cdot x$
Associative Property of Addition	$a+(b+c)=(a+b)+c$	We can group numbers in a sum any way we want and get the same answer.	$2+(3+5.1)=(2+3)+5.1$ $x+(5+7)=(x+5)+7$
Associative Property of Multiplication	$a(bc)=(ab)c$	We can group numbers in a product any way we want and get the same answer.	$2\cdot (3\cdot 4)=(2\cdot 3)\cdot 4$ $(-4y)\cdot x=-4(yx)$
Distributive Property	$a(b+c)=ab+ac$ $a(b-c)=ab-ac$	We can distribute multiplication over all terms of the sums or differences within parentheses	$2(3\pm 7)=2\cdot 3\pm 2\cdot 7$ $x(3\pm 5)=3x\pm 5x$
Additive identity property	$a+0=0+a=a$	Adding zero to any number yields the same number	$2+0=0+2=2$ $3x+0=0+3x=3x$
Multiplicative identity property	$a\cdot 1=1\cdot a=a$	Multiplying any number by 1 yields the same number	$3\cdot 1=1\cdot 3=3$ $1\cdot x=x\cdot 1=x$
Additive inverse property	$a+(-a)=0$	If we add a number and its opposite, we will get 0	$3+(-3)=0$ $2x+(-2x)=0$
Multiplicative inverse property	$a\cdot {\displaystyle \frac{1}{a}}=1(a\ne 0)$	If we multiply a nonzero number and its reciprocal, we will get 1	$3\cdot {\displaystyle \frac{1}{3}}=1$ $(x+1)\cdot {\displaystyle \frac{1}{x+1}}=1(x\ne -1)$