1. Classifying numbers.

   Natural numbers	$\mathbf{N}=\{1,2,3,\dots \}$
   Whole numbers	$\mathbf{W}=\{0,1,2,3,\dots \}$
   Integers	$\mathbf{Z}=\{\dots ,-2,-1,0,1,2,\dots \}$
   Rational numbers	Q: Numbers that can be expressed as the quotient ${\displaystyle \frac{a}{b}}$ of two integers, where $b\ne 0;$ a terminating or repeating decimal
   Irrational numbers	Decimals that neither terminate nor repeat
   Real numbers	R: All of the rational and irrational numbers

2. Coordinate line. Real numbers can be associated with a geometric line called a number line or coordinate line. Numbers associated with points to the right (left) of zero are called positive (negative) numbers.

3. Ordering numbers. $a<b$ means that $b=a+c$ for some positive number c.

   Trichotomy property. For two numbers a and b, exactly one of the following is true: $a<b,$ $a=b,$ or $a>b.$

   Transitive property. If $a<b$ and $b<c,$ then $a<c.$

4. Absolute value.

   $$|a|=a\text{if }a\ge 0,\text{ and }|\text{a}|=-a\text{if }a<0.$$

   The distance between points a and b on a number line is $|a-b|$.

5. Algebraic properties. Let a, b, and c be real numbers.

   Closure	$a+b$ is a real number, $a\cdot b$ is a real number.
   Commutative	$a+b=b+a,ab=ba$
   Associative	$\begin{array}{rcl}(a+b)+c& =& a+(b+c)\\ (ab)c& =& a(bc)\end{array}$
   Distributive	$a\cdot (b+c)=a\cdot b+a\cdot c$ $(a+b)\cdot c=a\cdot c+b\cdot c$
   Identity	$a+0=0+a=a,$$a\cdot 1=1\cdot a=a$
   Inverse properties	$a+(-a)=0$, $a\cdot {\displaystyle \frac{1}{a}}=1,$ where $a\ne 0$
   Zero-product properties	$0\cdot a=0=a\cdot 0$ If $a\cdot b=0,$ then $a=0$ or $b=0.$

1. If n is a positive integer, then

   $$\underset{n\text{ factros}}{\underbrace{a^n=a\cdot a\cdots a}}$$
   $$\begin{array}{l}{a}^0=1\\ a^{-n}={\displaystyle {\displaystyle \frac{1}{a^n}}}\end{array}\}a\ne 0$$

2. Rules of exponents. Denominators and exponents attached to a base of zero are assumed not to be zero.

   Product rule	$a^m\cdot a^n=a^{m+n}$
   Quotient rule	${\displaystyle \frac{a^m}{a^n}}=a^{m-n}$
   Power-of-a-power rule	${(a^m)}^n=a^{mn}$
   Power-of-a-product rule	${(a\cdot b)}^n=a^n\cdot b^n$
   Power-of-a-quotient rules	${\left({\displaystyle \frac{a}{b}}\right)}^n={\displaystyle \frac{a^n}{b^n}}$ ${\left({\displaystyle \frac{a}{b}}\right)}^{-n}={\left({\displaystyle \frac{b}{a}}\right)}^n={\displaystyle \frac{b^n}{a^n}}$

An algebraic expression of the form

where n is a nonnegative integer, is a polynomial in x. If $a_n\ne 0,$ then n is called the degree of the polynomial. The term $a_n{x}^n$ is the leading term, and $a_0$ is the constant term.

To add or subtract two polynomials, we add or subtract the coefficients of the like terms.

To multiply two polynomials, distribute each term of the first polynomial to each term of the second polynomial and then combine terms.

Special Products

Factoring formulas:

• $A^2-B^2=(A-B)(A+B)$

• $A^2+2AB+B^2={(A+B)}^2$

• $A^2-2AB+B^2={(A-B)}^2$

• $A^3-B^3=(A-B)(A^2+AB+B^2)$

• $A^3+B^3=(A+B)(A^2-AB+B^2)$

The quotient of two polynomials is called a rational expression.

1. Multiplication and division.

   $$\begin{array}{rcl}{\displaystyle {\displaystyle \frac{A}{B}}}\cdot {\displaystyle {\displaystyle \frac{C}{D}}}& =& {\displaystyle {\displaystyle \frac{A\cdot C}{B\cdot D}}}\\ {\displaystyle {\displaystyle \frac{{\displaystyle {\displaystyle \frac{A}{B}}}}{{\displaystyle {\displaystyle \frac{C}{D}}}}}}& =& {\displaystyle {\displaystyle \frac{A}{B}}}÷{\displaystyle {\displaystyle \frac{C}{D}}}={\displaystyle {\displaystyle \frac{A}{B}}}\cdot {\displaystyle {\displaystyle \frac{D}{C}}}={\displaystyle {\displaystyle \frac{A\cdot D}{B\cdot C}}}\end{array}$$

   All of the denominators are assumed not to be zero.

2. Addition and subtraction.

   $${\displaystyle {\displaystyle \frac{A}{B}}}\pm {\displaystyle {\displaystyle \frac{C}{D}}}={\displaystyle {\displaystyle \frac{A\cdot D\pm B\cdot C}{B\cdot D}}}$$

   The denominators are assumed not to be zero.

1. If $a>0,$ then $\sqrt[n]{a}=b$ means that $b^n=a$ and $b>0.$

2. If $a<0$ and n is odd, then $\sqrt[n]{a}=b$ means that $b^n=a$. If $a<0$ and n is even, then $\sqrt[n]{a}$ is not defined.

3. When all expressions are defined,

   $$a^{m/n}=\sqrt{n{a}^m}={(\sqrt[n]{a})}^m;a^{-\text{ m}/n}={\displaystyle {\displaystyle \frac{1}{a^{m/n}}}}.$$

4. $\sqrt{a^2}=|a|$