Preliminary Notions

Addition and Subtraction

Distributive Property

a(b + c) = ab + ac

(+) \cdot (+) = (+)

(-) \cdot (+) = (-)

(+) \cdot (-) = (-)

(-) \cdot (-) = (+)

Associative Property

a \cdot (b \cdot c) = (a \cdot b) \cdot c

a^{m} \cdot a^{n} = a^{m+n}

(a \cdot b)^{n} = a^{n} \cdot b^{n}

(a^m)^n = a^{m \cdot n}

(a^{\alpha} \cdot b^{\beta})^n = a^{\alpha n} \cdot b^{\beta n}

(a \pm b)^2 = a^2 \pm 2ab + b^2

(a+b)(a-b) = a^2 - b^2

(x+a)(x+b) = x^2 + (a+b)x + ab

\frac{(+)}{(+)} = (+)

\frac{(-)}{(+)} = (-)

\frac{(+)}{(-)} = (-)

\frac{(-)}{(-)} = (+)

Exponent Properties

\frac{a^{m}}{a^{n}} = a^{m-n} \quad (a \neq 0)

\left( \frac{a}{b} \right)^n = \frac{a^n}{b^n} \quad (b \neq 0)

\left( \frac{a^\alpha}{b^\beta} \right)^n = \frac{a^{\alpha n}}{b^{\beta n}} \quad (b \neq 0)

Inverses

a^{-n} = \frac{1}{a^n} \quad \text{; } a \neq 0

$0^{-n}$ is undefined for $n > 0$ .

Distributive Property

\frac{a + b}{c} = \frac{a}{c} + \frac{b}{c} \quad (c \neq 0)

Operations with Fractions
Condition: $y, z, w, k \neq 0$

\frac{x}{y} = x \left( \frac{1}{y} \right)

\left( \frac{x}{y} \right) \left( \frac{w}{k} \right) = \frac{xw}{yk}

\frac{xy}{wx} = \frac{y}{w}

\frac{x}{y} + \frac{z}{y} = \frac{x + z}{y}

\frac{x}{y} + \frac{w}{z} = \frac{xz + yw}{yz}

\frac{x}{y} \div \frac{w}{z} = \frac{xz}{yw}

x + \frac{y}{w} = \frac{xw + y}{w}

Division by zero is undefined. All denominators must be $\neq 0$ .

-\frac{a}{b} = \frac{-a}{b} = \frac{a}{-b}

a+\frac{b}{c+\dfrac{x}{y}}=a+\frac{b}{\dfrac{cy+x}{y}}=a+\frac{by}{cy+x} \quad (cy + x \neq 0)

Concept	Symbol	Concept	Symbol	Concept	Symbol
plus	+	greater than	$>$	there exists at least one	$\exists$
minus	-	less than	$<$	there exists exactly one	$\exists!$
multiplication	$\cdot$	greater than or equal to	$\geq$	there does not exist	$\nexists$
division	$\div$	less than or equal to	$\leq$	therefore	$\rightarrow$
equals	=	belongs to	$\in$	if and only if	$\leftrightarrow$
not equal	$\neq$	does not belong to	$\notin$	negation (not)	$\sim$
identical to	$\equiv$	subset or equal to	$\subseteq$	logical AND	$\land$
not identical to	$\not\equiv$	proper subset	$\subset$	logical OR	$\lor$
approximately equal	$\approx$	not a subset	$\not\subset$	set of natural numbers	$\mathbb{N}$
infinity	$\infty$	empty set	$\varnothing$	set of integers	$\mathbb{Z}$
positive infinity	$+\infty$	open interval $(a,b)$	$(a,b)$	set of rational numbers	$\mathbb{Q}$
negative infinity	$-\infty$	closed interval $[a,b]$	$[a,b]$	set of irrational numbers	$\mathbb{I}$
union	$\cup$	half-open interval $[a,b)$	$[a,b)$	set of real numbers	$\mathbb{R}$
intersection	$\cap$	real number line	$(-\infty, \infty)$	set of complex numbers	$\mathbb{C}$
therefore	$\therefore$	summation	$\sum$	factorial	$n!$
because	$\because$	product	$\prod$	absolute value of $x$	$\lvert x \rvert$
parallel	$\parallel$	square root	$\sqrt{}$	floor of $x$ (greatest integer ≤ $x$ )	$\lfloor x \rfloor$
not parallel	$\nparallel$	exponentiation	$a^b$	percentage	$\%$
such that	$\mid$	for all	$\forall$	multiple of $x$	$\dot{x}$