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 the same letters, for example by $R$.

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,...\} $

• The three dots, known as an ellipsis, signify that the pattern continues indefinitely.

The numbers $-1,-2,-3,...$ 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}=\{...,-3,-2,-1,0,1,2,3,...\} $

which can also be written as

$ \mathbb{Z}=\{0,\pm1,\pm2,\pm3,\pm4,...\}. $

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 $0.3333...$ is the same as $1/3$).
The set of all rational numbers is denoted by $\mathbb{Q}$

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

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 $\pi=3.141592....$ Such numbers are called irrational numbers. The decimal digit representation
of an irrational number goes on forever and never repeats. 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}. $

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}\}. $

Why is $\sqrt{2}$ irrational?

To prove that \( \sqrt{2} \) is irrational, we can use a classic proof by contradiction. *Proof.* - **Assume the Opposite:** Suppose \( \sqrt{2} \) is rational. Then it can be expressed as a fraction \( \frac{p}{q} \), where \( p \) and \( q \) are integers with no common factors (i.e., the fraction is in simplest form), and \( q \neq 0 \). - **Square Both Sides:** Since \( \sqrt{2} = \frac{p}{q} \), squaring both sides gives: $ 2 = \frac{p^2}{q^2} $ - **Rewrite the Equation:** Multiply both sides by \( q^2 \) to eliminate the denominator: $ p^2 = 2q^2 $ This equation shows that \( p^2 \) is an even number because it is equal to \( 2 \times q^2 \). - **Conclude \( p \) is Even:** If \( p^2 \) is even, then \( p \) must also be even (because the square of an odd number is odd). Therefore, we can write \( p = 2k \) for some integer \( k \). - **Substitute \( p = 2k \) into the Equation:** Replace \( p \) in the equation \( p^2 = 2q^2 \): $ (2k)^2 = 2q^2 $ $ 4k^2 = 2q^2 $ $ 2k^2 = q^2 $ This equation shows that \( q^2 \) is also even, which means that \( q \) must be even as well. - **Contradiction:** If both \( p \) and \( q \) are even, then they have a common factor of 2. However, this contradicts our original assumption that \( \frac{p}{q} \) is in simplest form with no common factors. **Conclusion:** Since assuming that \( \sqrt{2} \) is rational leads to a contradiction, we conclude that \( \sqrt{2} \) must be irrational. $\square$

Geometric Interpretation of Real Numbers as Points on a Line

Real numbers can be visualized on a straight line called the **real number line** or **real line**. Each real number corresponds to a unique point on the line, and each point on the real line corresponds to a unique number.

The geometric representation of real numbers as points on a straight line is a familiar concept. In this representation, a specific point is designated as 0, while another point to its right is chosen to represent 1, as illustrated below. This selection establishes the scale for the line. Positive numbers lie to the right of the origin (0), while negative numbers lie to the left. The point twice the distance from 0 to 1 is labeled 2, while the point the same distance to the left of 0 is labeled $-1$, and so on. This way, every real number corresponds to a unique point on the line, and conversely, every point on the line corresponds to a unique real number, called its **coordinate**. Because of this one-to-one correspondence, the line is often referred to as the **real number line** or the **real line**, and it's customary to use the terms "real number" and "point" interchangeably. We often say "the point $x$" instead of specifying "the point corresponding to the real number $x$."

Basic Rules of Algebra

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.

|p{2.5cm}|p{4cm}|p{4cm}|} Name	Math	Description	Example \tabularnewline Commutative Property of Addition	$a+b=b+a$	We can add numbers in any order	**$2+3=3+2$ $ 4+2x=2x+4 $ **\tabularnewline Commutative Property of Multiplication	$ab=ba$	We can multiply in any order	**$2\cdot3=3\cdot2$ $ x\cdot4=4\cdot x $ **\tabularnewline 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.	**$2+(3+5)=(2+3)+5$ $ x+(5+7)=(x+5)+7 $ **\tabularnewline Associative Property of Multiplication	$a(bc)=(ab)c$	We can group numbers in a product any way we want and get the same answer.	**$2\cdot(3\cdot4)=(2\cdot3)\cdot4$ $ (-4y)\cdot x=-4(yx) $ **\tabularnewline Distributive Property	$a(b+c)=ab+ac$ $ a(b-c)=ab-bc $	We can distribute multiplication over all terms of the sums or differences within parentheses	{$2\cdot(3\pm7)=2\cdot3\pm2\cdot7$ $ x\cdot(3\pm5)=3x\pm5x $ }\tabularnewline 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 $ **\tabularnewline Multiplicative Identity Property	$a\cdot1=1\cdot a=a$	Multiplying any number by 1 yields the same number	**$3\cdot1=1\cdot3=3$ $ 1\cdot x=x $ **\tabularnewline Additive Inverse Property	$a+(-a)=0$	If we add a number and its opposite, we will get 0	**$3+(-3)=0$ $ 2x+(-2x)=0 $ **\tabularnewline Multiplicative Inverse Property	$a\cdot\dfrac{1}{a}=1\quad(a\neq0)$	If we multiply a nonzero number and its reciprocal, we will get 1	**$3\cdot\dfrac{1}{3}=1$ $ (x+1)\cdot\dfrac{1}{x+1}=1 $$ (\text{if }x\neq-1) $ **\tabularnewline

Here are some properties of real numbers:

Proof. We know

$ \begin{aligned} a=a\cdot 1&=a\cdot (1+0) &=a\cdot 1 + a\cdot 0 & = a +a\cdot 0 \end{aligned} $

Comparing $a=a+a\cdot 0$ and $a=a+0$, we conclude that $a\cdot 0 =0$. $\square$

• Note that $0$ has no multiplicative inverse because if \( 1/0 = b \), it would imply \( 1 = 0 \cdot b \), but \( 0 \cdot b = 0 \neq 1 \). This demonstrates that we cannot assign a definite value to \( 1/0 \). Therefore, division by zero is not a valid operation, and if \( a \) is any number, \( a/0 \) has no meaning.
• If $a\neq0$, then $\dfrac{0}{a}=0$ because $a\cdot0=0$. (again
  $a/0$ has no meaning).

Proof. We consider two cases. If $a=0$ then there is nothing more to prove. However, if $a\neq 0$, then $a^{-1}$ exists and we can multiply both sides of the equation by $a^{-1}$, giving $b=0\cdot a^{-1}=0$. $\square$