Math Types & Symbols

Cheatsheet Content

### Number Sets - **Natural Numbers** ($\mathbb{N}$): $\{1, 2, 3, ...\}$ (sometimes includes 0: $\{0, 1, 2, ...\}$) - **Integers** ($\mathbb{Z}$): $\{..., -2, -1, 0, 1, 2, ...\}$ - **Rational Numbers** ($\mathbb{Q}$): Numbers that can be expressed as a fraction $p/q$ where $p, q \in \mathbb{Z}$ and $q \neq 0$. E.g., $1/2, -3, 0.75$. - **Real Numbers** ($\mathbb{R}$): All rational and irrational numbers. Can be represented on a continuous number line. E.g., $\pi, \sqrt{2}, -5$. - **Complex Numbers** ($\mathbb{C}$): Numbers of the form $a + bi$, where $a, b \in \mathbb{R}$ and $i$ is the imaginary unit ($i^2 = -1$). ### Basic Arithmetic Operations | Symbol | Name | Example | Definition | |--------|----------------|----------------|------------------------------------------| | $+$ | Addition | $a+b$ | Sum of $a$ and $b$ | | $-$ | Subtraction | $a-b$ | Difference between $a$ and $b$ | | $\times$ or $\cdot$ | Multiplication | $a \times b$ or $a \cdot b$ | Product of $a$ and $b$ | | $\div$ or $/$ | Division | $a \div b$ or $a/b$ | Quotient of $a$ by $b$ ($b \neq 0$) | | $=$ | Equality | $a=b$ | $a$ is equal to $b$ | | $\neq$ | Inequality | $a \neq b$ | $a$ is not equal to $b$ | | $ $ | Greater than | $a > b$ | $a$ is strictly greater than $b$ | | $\le$ | Less than or equal to | $a \le b$ | $a$ is less than or equal to $b$ | | $\ge$ | Greater than or equal to | $a \ge b$ | $a$ is greater than or equal to $b$ | ### Algebraic Symbols - **Variable**: A symbol (usually a letter like $x, y, z$) representing an unknown quantity. - **Constant**: A fixed value that does not change. E.g., $5, \pi, e$. - **Coefficient**: A numerical or constant quantity placed before and multiplying a variable in an algebraic expression (e.g., 3 in $3x$). - **Exponent**: $a^n$ means $a$ multiplied by itself $n$ times. - **Radical/Root**: $\sqrt{x}$ (square root), $\sqrt[n]{x}$ (nth root). - **Absolute Value**: $|x|$ denotes the distance of $x$ from zero on the number line, always non-negative. - **Parentheses**: $( ), [ ], \{ \}$ used for grouping expressions and defining order of operations. ### Set Theory - **Set**: A collection of distinct objects, e.g., $A = \{1, 2, 3\}$. - **Element of**: $x \in A$ means $x$ is an element of set $A$. - **Not an element of**: $x \notin A$ means $x$ is not an element of set $A$. - **Subset**: $A \subseteq B$ means every element of $A$ is also an element of $B$. - **Proper Subset**: $A \subset B$ means $A \subseteq B$ and $A \neq B$. - **Union**: $A \cup B$ contains all elements that are in $A$, or in $B$, or in both. - **Intersection**: $A \cap B$ contains all elements that are in both $A$ and $B$. - **Empty Set**: $\emptyset$ or {} is the set containing no elements. - **Universal Set**: $U$ is the set of all possible elements under consideration. - **Complement**: $A^c$ or $A'$ contains all elements in $U$ that are not in $A$. - **Cardinality**: $|A|$ denotes the number of elements in set $A$. ### Logic Symbols | Symbol | Name | Meaning | Example | |--------|-----------------------|------------------------------------------|-------------------| | $\land$ | Conjunction (AND) | True if both $P$ and $Q$ are true | $P \land Q$ | | $\lor$ | Disjunction (OR) | True if $P$ is true, or $Q$ is true, or both | $P \lor Q$ | | $\neg$ | Negation (NOT) | True if $P$ is false | $\neg P$ | | $\implies$ | Implication (If...then) | If $P$ is true, then $Q$ is true | $P \implies Q$ | | $\iff$ | Biconditional (If and only if) | $P$ is true if and only if $Q$ is true | $P \iff Q$ | | $\forall$ | Universal Quantifier (For all) | For every element in a set | $\forall x \in S$ | | $\exists$ | Existential Quantifier (There exists) | There is at least one element in a set | $\exists x \in S$ | ### Geometry Symbols - **Line**: $\overleftrightarrow{AB}$ - **Line Segment**: $\overline{AB}$ - **Ray**: $\overrightarrow{AB}$ - **Angle**: $\angle ABC$ - **Perpendicular**: $\perp$ (e.g., $L_1 \perp L_2$) - **Parallel**: $\parallel$ (e.g., $L_1 \parallel L_2$) - **Triangle**: $\triangle ABC$ - **Degree**: $^\circ$ (e.g., $90^\circ$) - **Pi**: $\pi \approx 3.14159$ (ratio of a circle's circumference to its diameter) - **Congruent**: $\cong$ (same shape and size) - **Similar**: $\sim$ (same shape, different size) ### Calculus Symbols - **Limit**: $\lim_{x \to a} f(x)$ (the value $f(x)$ approaches as $x$ approaches $a$) - **Derivative**: $\frac{dy}{dx}$ or $f'(x)$ (rate of change of $y$ with respect to $x$) - **Partial Derivative**: $\frac{\partial f}{\partial x}$ (derivative of a multi-variable function with respect to one variable) - **Integral**: $\int f(x) dx$ (antiderivative or area under a curve) - **Definite Integral**: $\int_a^b f(x) dx$ (integral from $a$ to $b$) - **Summation**: $\sum_{i=1}^n a_i$ (sum of $a_i$ from $i=1$ to $n$) - **Infinity**: $\infty$ (a concept representing something endless or unbounded) - **Approximation**: $\approx$ (approximately equal to) ### Functions - **Function Notation**: $f(x)$ (a function named $f$ with input $x$) - **Domain**: The set of all possible input values for which a function is defined. - **Range**: The set of all possible output values of a function. - **Inverse Function**: $f^{-1}(x)$ (reverses the action of $f$) - **Composition of Functions**: $(f \circ g)(x) = f(g(x))$ - **Exponential Function**: $e^x$ (where $e \approx 2.71828$ is Euler's number) - **Logarithm**: $\log_b x$ (the power to which $b$ must be raised to get $x$) - **Natural Logarithm**: $\ln x = \log_e x$ - **Common Logarithm**: $\log x = \log_{10} x$ ### Linear Algebra - **Vector**: $\vec{v}$ or $\mathbf{v}$ (a quantity having magnitude and direction) - **Matrix**: $\mathbf{A}$ (a rectangular array of numbers) - **Determinant**: $\det(\mathbf{A})$ or $|\mathbf{A}|$ (a scalar value associated with a square matrix) - **Identity Matrix**: $\mathbf{I}$ (a square matrix with ones on the main diagonal and zeros elsewhere) - **Transpose**: $\mathbf{A}^T$ (rows become columns and columns become rows) - **Inverse Matrix**: $\mathbf{A}^{-1}$ (such that $\mathbf{A}\mathbf{A}^{-1} = \mathbf{I}$) - **Eigenvalue**: $\lambda$ (a scalar such that $A\mathbf{v} = \lambda\mathbf{v}$) - **Eigenvector**: $\mathbf{v}$ (a non-zero vector that changes at most by a scalar factor when a linear transformation is applied to it) ### Statistics & Probability - **Probability of Event A**: $P(A)$ - **Mean (Average)**: $\bar{x}$ or $\mu$ (sum of values divided by count) - **Median**: The middle value in an ordered dataset. - **Mode**: The value that appears most often in a dataset. - **Standard Deviation**: $\sigma$ or $s$ (measure of spread of data) - **Variance**: $\sigma^2$ or $s^2$ (square of standard deviation) - **Factorial**: $n! = n \times (n-1) \times ... \times 1$ - **Combination**: $C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}$ (number of ways to choose $k$ items from $n$ without regard to order) - **Permutation**: $P(n, k) = \frac{n!}{(n-k)!}$ (number of ways to choose $k$ items from $n$ with regard to order) ### Greek Alphabet (Commonly Used in Math) | Uppercase | Lowercase | Name | Common Usage Examples | |-----------|-----------|-----------|------------------------------------------------| | $\Alpha$ | $\alpha$ | Alpha | Angles, coefficients, significance level | | $\Beta$ | $\beta$ | Beta | Angles, coefficients | | $\Gamma$ | $\gamma$ | Gamma | Angles, Euler-Mascheroni constant | | $\Delta$ | $\delta$ | Delta | Change, increment, variation | | $\Epsilon$ | $\epsilon$ | Epsilon | Small positive quantity (limits), error terms | | $\Zeta$ | $\zeta$ | Zeta | Riemann zeta function | | $\Eta$ | $\eta$ | Eta | Efficiency, learning rate | | $\Theta$ | $\theta$ | Theta | Angles | | $\Lambda$ | $\lambda$ | Lambda | Eigenvalues, wavelength, Lagrange multiplier | | $\Mu$ | $\mu$ | Mu | Mean, population mean, coefficient of friction | | $\Pi$ | $\pi$ | Pi | Ratio of circumference to diameter, product | | $\Rho$ | $\rho$ | Rho | Density, correlation coefficient | | $\Sigma$ | $\sigma$ | Sigma | Standard deviation, sum (capital) | | $\Tau$ | $\tau$ | Tau | Torque, time constant | | $\Phi$ | $\phi$ | Phi | Golden ratio, angles | | $\Chi$ | $\chi$ | Chi | Chi-squared distribution | | $\Psi$ | $\psi$ | Psi | Wave function | | $\Omega$ | $\omega$ | Omega | Angular velocity, last element |