📝 Notation Cheatsheet

Essential mathematical symbols, notation, and terminology for MATHCOUNTS success.

Basic Operations

Arithmetic Symbols

• $+$ : Addition
• $-$ : Subtraction
• $\times$ or $\cdot$ : Multiplication
• $\div$ or $/$ : Division
• $=$ : Equals
• $\neq$ : Not equal
• $\approx$ : Approximately equal
• $>$ : Greater than
• $<$ : Less than
• $\geq$ : Greater than or equal
• $\leq$ : Less than or equal

Powers and Roots

• $a^2$ : $a$ squared
• $a^3$ : $a$ cubed
• $a^n$ : $a$ to the $n$th power
• $\sqrt{a}$ : Square root of $a$
• $\sqrt[n]{a}$ : $n$th root of $a$
• $a^{1/n}$ : $n$th root of $a$ (alternative notation)

Fractions and Decimals

Fraction Notation

• $\frac{a}{b}$ : Fraction $a$ over $b$
• $\frac{a}{b} = \frac{c}{d}$ : Proportion
• $\frac{a}{b} \pm \frac{c}{d}$ : Adding/subtracting fractions
• $\frac{a}{b} \times \frac{c}{d}$ : Multiplying fractions
• $\frac{a}{b} \div \frac{c}{d}$ : Dividing fractions

Decimal Notation

• $0.5$ : Five tenths
• $0.25$ : Twenty-five hundredths
• $0.333\ldots$ : Repeating decimal
• $0.\overline{3}$ : Repeating decimal (bar notation)
• $1.23 \times 10^4$ : Scientific notation

Percent Notation

• $50%$ : Fifty percent
• $0.5 = 50%$ : Decimal to percent
• $\frac{1}{2} = 50%$ : Fraction to percent
• $x%$ : $x$ percent

Sets and Logic

Set Notation

• ${1, 2, 3}$ : Set containing 1, 2, 3
• $\emptyset$ : Empty set
• $\in$ : Element of
• $\notin$ : Not an element of
• $\subset$ : Subset of
• $\cap$ : Intersection
• $\cup$ : Union
• $A’$ : Complement of set $A$

Logic Symbols

• $\land$ : And
• $\lor$ : Or
• $\neg$ : Not
• $\Rightarrow$ : Implies
• $\Leftrightarrow$ : If and only if
• $\forall$ : For all
• $\exists$ : There exists

Geometry

Angle Notation

• $\angle ABC$ : Angle with vertex $B$
• $\angle 1$ : Angle 1
• $m\angle ABC$ : Measure of angle $ABC$
• $90°$ : 90 degrees
• $\pi$ : Pi (approximately 3.14159)
• $180° = \pi$ radians

Triangle Notation

• $\triangle ABC$ : Triangle with vertices $A$, $B$, $C$
• $AB$ : Line segment from $A$ to $B$
• $\overline{AB}$ : Line segment from $A$ to $B$
• $|AB|$ : Length of segment $AB$
• $AB \parallel CD$ : $AB$ is parallel to $CD$
• $AB \perp CD$ : $AB$ is perpendicular to $CD$

Circle Notation

• $O$ : Center of circle
• $r$ : Radius
• $d$ : Diameter
• $C$ : Circumference
• $A$ : Area
• $\widehat{AB}$ : Arc from $A$ to $B$

Algebra

Variable Notation

• $x, y, z$ : Variables
• $a, b, c$ : Constants
• $f(x)$ : Function of $x$
• $f’(x)$ : Derivative of $f(x)$ (advanced)
• $f^{-1}(x)$ : Inverse function of $f(x)$

Function Notation

• $f(x) = 2x + 3$ : Linear function
• $f(x) = x^2$ : Quadratic function
• $f(x) = \sqrt{x}$ : Square root function
• $f(x) = |x|$ : Absolute value function

Equation Types

• $ax + b = 0$ : Linear equation
• $ax^2 + bx + c = 0$ : Quadratic equation
• $ax + by = c$ : Linear equation in two variables
• $x^2 + y^2 = r^2$ : Circle equation

Statistics and Probability

Statistical Notation

• $\bar{x}$ : Mean (average)
• $x_1, x_2, \ldots, x_n$ : Data values
• $n$ : Sample size
• $P(A)$ : Probability of event $A$
• $P(A|B)$ : Probability of $A$ given $B$

Probability Symbols

• $P(A)$ : Probability of event $A$
• $P(A \cap B)$ : Probability of $A$ and $B$
• $P(A \cup B)$ : Probability of $A$ or $B$
• $P(A’)$ : Probability of not $A$
• $P(A|B)$ : Probability of $A$ given $B$

Sequences and Series

Sequence Notation

• $a_n$ : $n$th term of sequence
• $a_1, a_2, a_3, \ldots$ : Sequence terms
• $S_n$ : Sum of first $n$ terms
• $a_{n+1} = a_n + d$ : Arithmetic sequence
• $a_{n+1} = a_n \cdot r$ : Geometric sequence

Series Notation

• $\sum_{i=1}^{n} a_i$ : Sum from $i=1$ to $n$
• $\sum_{i=1}^{\infty} a_i$ : Infinite series
• $S_n = \frac{n}{2}(a_1 + a_n)$ : Sum of arithmetic series
• $S_n = \frac{a_1(1-r^n)}{1-r}$ : Sum of geometric series

Units and Measurements

Length Units

• $m$ : Meter
• $cm$ : Centimeter
• $mm$ : Millimeter
• $km$ : Kilometer
• $in$ : Inch
• $ft$ : Foot
• $yd$ : Yard
• $mi$ : Mile

Area Units

• $m^2$ : Square meter
• $cm^2$ : Square centimeter
• $ft^2$ : Square foot
• $in^2$ : Square inch
• $acre$ : Acre

Volume Units

• $m^3$ : Cubic meter
• $cm^3$ : Cubic centimeter
• $ft^3$ : Cubic foot
• $in^3$ : Cubic inch
• $L$ : Liter
• $mL$ : Milliliter
• $gal$ : Gallon
• $qt$ : Quart
• $pt$ : Pint

Time Units

• $s$ : Second
• $min$ : Minute
• $h$ : Hour
• $day$ : Day
• $week$ : Week
• $month$ : Month
• $year$ : Year

Common Abbreviations

Mathematical Terms

• $GCD$ : Greatest Common Divisor
• $LCM$ : Least Common Multiple
• $GCF$ : Greatest Common Factor
• $EV$ : Expected Value
• $PIE$ : Principle of Inclusion-Exclusion
• $RTD$ : Rate-Time-Distance
• $SAS$ : Side-Angle-Side
• $SSS$ : Side-Side-Side
• $ASA$ : Angle-Side-Angle
• $AAS$ : Angle-Angle-Side

Problem Types

• $AMC$ : American Mathematics Competition
• $AIME$ : American Invitational Mathematics Examination
• $USAMO$ : United States of America Mathematical Olympiad
• $IMO$ : International Mathematical Olympiad

Quick Reference Tables

Common Conversions

Common Fractions

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