Preliminary Notions

Addition and Subtraction

Tip

Distributive Property

a(b+c)=ab+aca(b + c) = ab + ac

Multiplication

Sign Rules

Tip

(+)⋅(+)=(+)(+) \cdot (+) = (+)

(−)⋅(+)=(−)(-) \cdot (+) = (-)

(+)⋅(−)=(−)(+) \cdot (-) = (-)

(−)⋅(−)=(+)(-) \cdot (-) = (+)

Tip

Associative Property

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

Exponent Properties

am⋅an=am+na^{m} \cdot a^{n} = a^{m+n}

(a⋅b)n=an⋅bn(a \cdot b)^{n} = a^{n} \cdot b^{n}

(am)n=am⋅n(a^m)^n = a^{m \cdot n}

(aα⋅bβ)n=aαn⋅bβn(a^{\alpha} \cdot b^{\beta})^n = a^{\alpha n} \cdot b^{\beta n}

Notable Equivalences

(a±b)2=a2±2ab+b2(a \pm b)^2 = a^2 \pm 2ab + b^2

(a+b)(a−b)=a2−b2(a+b)(a-b) = a^2 - b^2

(x+a)(x+b)=x2+(a+b)x+ab(x+a)(x+b) = x^2 + (a+b)x + ab

Division

Sign Rules

Tip

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

(−)(+)=(−)\frac{(-)}{(+)} = (-)

(+)(−)=(−)\frac{(+)}{(-)} = (-)

(−)(−)=(+)\frac{(-)}{(-)} = (+)

Exponent Properties

aman=am−n(a≠0)\frac{a^{m}}{a^{n}} = a^{m-n} \quad (a \neq 0)

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

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

Fundamental Theorems

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Inverses

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

0−n0^{-n} is undefined for n>0n > 0.

Tip

Distributive Property

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

Operations with Fractions

Condition: y,z,w,k≠0y, z, w, k \neq 0

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

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

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

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

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

xy÷wz=xzyw\frac{x}{y} \div \frac{w}{z} = \frac{xz}{yw}

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

Important Notes

Key Restriction

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

Useful Equivalences

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

a+bc+xy=a+bcy+xy=a+bycy+x(cy+x≠0)a+\frac{b}{c+\dfrac{x}{y}}=a+\frac{b}{\dfrac{cy+x}{y}}=a+\frac{by}{cy+x} \quad (cy + x \neq 0)

Mathematical Symbols

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 N\mathbb{N}
infinity ∞\infty empty set ∅\varnothing set of integers Z\mathbb{Z}
positive infinity +∞+\infty open interval (a,b)(a,b) (a,b)(a,b) set of rational numbers Q\mathbb{Q}
negative infinity −∞-\infty closed interval [a,b][a,b] [a,b][a,b] set of irrational numbers I\mathbb{I}
union ∪\cup half-open interval [a,b)[a,b) [a,b)[a,b) set of real numbers R\mathbb{R}
intersection ∩\cap real number line (−∞,∞)(-\infty, \infty) set of complex numbers C\mathbb{C}
therefore ∴\therefore summation ∑\sum factorial n!n!
because ∵\because product ∏\prod absolute value of xx ∣x∣\lvert x \rvert
parallel ∥\parallel square root \sqrt{} floor of xx (greatest integer ≤ xx) ⌊x⌋\lfloor x \rfloor
not parallel ∦\nparallel exponentiation aba^b percentage %\%
such that ∣\mid for all ∀\forall multiple of xx x˙\dot{x}