Math Fundamentals Cheatsheet

Cheatsheet Content

### Numbers & Operations - **Natural Numbers:** $1, 2, 3, ...$ (counting numbers) - **Whole Numbers:** $0, 1, 2, 3, ...$ (natural numbers + zero) - **Integers:** $..., -2, -1, 0, 1, 2, ...$ (whole numbers + negative counterparts) - **Rational Numbers:** Numbers that can be expressed as a fraction $\frac{p}{q}$ where $p, q$ are integers and $q \neq 0$. E.g., $\frac{1}{2}, -3, 0.75$. - **Irrational Numbers:** Numbers that cannot be expressed as a simple fraction. E.g., $\pi, \sqrt{2}$. - **Real Numbers:** All rational and irrational numbers. - **Operations:** - **Addition (+):** Combining quantities. $a + b$ - **Subtraction (-):** Finding the difference. $a - b$ - **Multiplication (× or *):** Repeated addition. $a \times b$ - **Division (÷ or /):** Splitting into equal parts. $a \div b$ ($b \neq 0$) - **Order of Operations (PEMDAS/BODMAS):** 1. **P**arentheses / **B**rackets 2. **E**xponents / **O**rders 3. **M**ultiplication and **D**ivision (from left to right) 4. **A**ddition and **S**ubtraction (from left to right) - Example: $5 + 2 \times (8 - 3)^2 = 5 + 2 \times (5)^2 = 5 + 2 \times 25 = 5 + 50 = 55$ ### Number Properties - **Commutative Property:** - Addition: $a + b = b + a$ (e.g., $3 + 5 = 5 + 3$) - Multiplication: $a \times b = b \times a$ (e.g., $3 \times 5 = 5 \times 3$) - **Associative Property:** - Addition: $(a + b) + c = a + (b + c)$ (e.g., $(1+2)+3 = 1+(2+3)$) - Multiplication: $(a \times b) \times c = a \times (b \times c)$ (e.g., $(1 \times 2) \times 3 = 1 \times (2 \times 3)$) - **Distributive Property:** $a \times (b + c) = (a \times b) + (a \times c)$ (e.g., $2 \times (3+4) = (2 \times 3) + (2 \times 4)$) - **Identity Property:** - Addition: $a + 0 = a$ (0 is the additive identity) - Multiplication: $a \times 1 = a$ (1 is the multiplicative identity) - **Inverse Property:** - Addition: $a + (-a) = 0$ ($-a$ is the additive inverse) - Multiplication: $a \times \frac{1}{a} = 1$ (for $a \neq 0$, $\frac{1}{a}$ is the multiplicative inverse) ### Fractions, Decimals & Percent - **Fractions:** Represents a part of a whole. $\frac{\text{Numerator}}{\text{Denominator}}$ - **Equivalent Fractions:** $\frac{1}{2} = \frac{2}{4} = \frac{5}{10}$ - **Adding/Subtracting:** Find a common denominator. $\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}$ - **Multiplying:** Multiply numerators and denominators. $\frac{1}{2} \times \frac{3}{4} = \frac{1 \times 3}{2 \times 4} = \frac{3}{8}$ - **Dividing:** Multiply by the reciprocal of the second fraction. $\frac{1}{2} \div \frac{3}{4} = \frac{1}{2} \times \frac{4}{3} = \frac{4}{6} = \frac{2}{3}$ - **Decimals:** A way to represent fractions where the denominator is a power of 10. - **Place Values:** ...hundreds, tens, ones . tenths, hundredths, thousandths... - **Conversion:** - Fraction to Decimal: Divide numerator by denominator. $\frac{3}{4} = 3 \div 4 = 0.75$ - Decimal to Fraction: $0.25 = \frac{25}{100} = \frac{1}{4}$ - **Percent (%):** "Per hundred." A ratio out of 100. - **Conversion:** - Decimal to Percent: Multiply by 100. $0.75 = 0.75 \times 100\% = 75\%$ - Percent to Decimal: Divide by 100. $25\% = 25 \div 100 = 0.25$ - Fraction to Percent: Convert to decimal then to percent. $\frac{1}{4} = 0.25 = 25\%$ ### Measurement - **Units:** - **Length:** meter (m), kilometer (km), centimeter (cm), millimeter (mm), inch (in), foot (ft), yard (yd), mile (mi) - **Mass/Weight:** gram (g), kilogram (kg), milligram (mg), pound (lb), ounce (oz) - **Volume:** liter (L), milliliter (mL), cubic meter ($m^3$), gallon (gal), quart (qt), pint (pt), cup (c) - **Time:** second (s), minute (min), hour (hr), day, year - **Temperature:** Celsius ($^\circ C$), Fahrenheit ($^\circ F$), Kelvin (K) - **Conversions:** Use conversion factors. E.g., $1 \text{ m} = 100 \text{ cm}$, $1 \text{ kg} = 1000 \text{ g}$ - To convert from a larger unit to a smaller unit, multiply. - To convert from a smaller unit to a larger unit, divide. - Example: Convert 2.5 meters to centimeters: $2.5 \text{ m} \times 100 \frac{\text{cm}}{\text{m}} = 250 \text{ cm}$ ### Geometry Basics - **Basic Shapes:** - **Point:** A location in space, no size. - **Line:** Extends infinitely in two directions. - **Line Segment:** Part of a line with two endpoints. - **Ray:** Part of a line with one endpoint, extending infinitely in one direction. - **Angle:** Formed by two rays sharing a common endpoint (vertex). - **Acute:** $ 90^\circ$ and $ ### Patterns & Pre-Algebra - **Patterns:** Recognizing sequences and rules. - **Arithmetic Sequence:** Common difference between terms. E.g., $2, 5, 8, 11, ...$ (add 3) - **Geometric Sequence:** Common ratio between terms. E.g., $2, 4, 8, 16, ...$ (multiply by 2) - **Variables:** Letters representing unknown values (e.g., $x, y, a$). - **Expressions:** Combinations of numbers, variables, and operations (e.g., $2x + 5$). - **Equations:** Statements that two expressions are equal (e.g., $2x + 5 = 11$). - **Solving Equations:** Isolating the variable using inverse operations. - Example: $2x + 5 = 11$ - Subtract 5 from both sides: $2x = 6$ - Divide by 2: $x = 3$ - **Inequalities:** Comparisons between expressions using $ , \le, \ge$. - Example: $x + 3 ### Data & Probability - **Data Representation:** - **Charts:** Bar charts, pie charts, line graphs. - **Tables:** Organizing data in rows and columns. - **Measures of Central Tendency:** - **Mean (Average):** Sum of all values divided by the number of values. - **Median:** The middle value when data is ordered. If even number of values, average of the two middle values. - **Mode:** The value that appears most frequently. - **Measures of Spread:** - **Range:** Difference between the highest and lowest values. - **Probability:** The likelihood of an event occurring. - $P(\text{Event}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$ - Probability values are between 0 and 1 (or 0% and 100%). - Example: Rolling a 3 on a standard 6-sided die. $P(3) = \frac{1}{6}$ ### Algebra Foundations - **Exponents:** $a^n = a \times a \times ... \times a$ ($n$ times) - **Rules:** - $a^m \times a^n = a^{m+n}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $(a^m)^n = a^{mn}$ - $(ab)^n = a^n b^n$ - $a^0 = 1$ (for $a \neq 0$) - $a^{-n} = \frac{1}{a^n}$ - **Radicals (Roots):** $\sqrt[n]{a}$ is a number that, when multiplied by itself $n$ times, equals $a$. - $\sqrt{a} = a^{1/2}$ (square root) - $\sqrt[3]{a} = a^{1/3}$ (cube root) - **Polynomials:** Expressions with one or more terms, where each term is a constant multiplied by one or more variables raised to non-negative integer powers. - **Monomial:** $3x^2$ - **Binomial:** $x+y$ - **Trinomial:** $x^2 + 2x + 1$ - **Adding/Subtracting:** Combine like terms. $(2x+3) + (x-1) = 3x+2$ - **Multiplying:** Use distributive property (FOIL for binomials). $(x+1)(x+2) = x^2+2x+x+2 = x^2+3x+2$ - **Factoring:** The reverse of multiplying polynomials. - **Greatest Common Factor (GCF):** $3x^2 + 6x = 3x(x+2)$ - **Difference of Squares:** $a^2 - b^2 = (a-b)(a+b)$ - **Trinomials:** $x^2 + bx + c = (x+p)(x+q)$ where $p+q=b, pq=c$. E.g., $x^2+5x+6=(x+2)(x+3)$ - **Rational Expressions:** Fractions where numerator and/or denominator are polynomials. - Simplify by factoring and canceling common factors. $\frac{x^2-1}{x-1} = \frac{(x-1)(x+1)}{x-1} = x+1$ (for $x \neq 1$) ### Functions Basics - **Definition:** A relation where each input (domain) has exactly one output (range). - Represented as $f(x) = \text{expression}$ - **Domain:** All possible input values ($x$). - **Range:** All possible output values ($y$ or $f(x)$). - **Evaluating Functions:** Substitute input value into the expression. - If $f(x) = 2x + 1$, then $f(3) = 2(3) + 1 = 7$. - **Graphing Functions:** Plotting input-output pairs $(x, f(x))$ on a coordinate plane. - **Linear Functions:** $y = mx + b$ (straight line) - $m$: slope (rate of change) - $b$: y-intercept (where line crosses y-axis) - **Quadratic Functions:** $y = ax^2 + bx + c$ (parabola) - Vertex: $(-\frac{b}{2a}, f(-\frac{b}{2a}))$ - **Inverse Functions:** Function that "undoes" the original function. $f^{-1}(f(x)) = x$. - To find: Swap $x$ and $y$, then solve for $y$. ### Geometry (Advanced) - **Pythagorean Theorem:** For a right-angled triangle, $a^2 + b^2 = c^2$, where $c$ is the hypotenuse. - **Types of Triangles:** - **Equilateral:** All sides equal, all angles $60^\circ$. - **Isosceles:** Two sides equal, two angles equal. - **Scalene:** All sides different, all angles different. - **Congruence:** Shapes are congruent if they have the same size and shape. - **SSS, SAS, ASA, AAS** congruence postulates for triangles. - **Similarity:** Shapes are similar if they have the same shape but different sizes (angles are equal, sides are proportional). - **Transformations:** - **Translation:** Sliding a shape. - **Rotation:** Turning a shape around a point. - **Reflection:** Flipping a shape over a line. - **Dilation:** Resizing a shape. ### Basic Trigonometry - **Right Triangle Ratios (SOH CAH TOA):** For an acute angle $\theta$ in a right triangle: - $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$ - $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$ - $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$ - **Reciprocal Ratios:** - $\csc(\theta) = \frac{1}{\sin(\theta)}$ - $\sec(\theta) = \frac{1}{\cos(\theta)}$ - $\cot(\theta) = \frac{1}{\tan(\theta)}$ - **Unit Circle:** A circle with radius 1 centered at the origin, used to define trig functions for all angles. - $(x,y) = (\cos\theta, \sin\theta)$ - **Special Angles:** Know values for $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$ (and their radian equivalents). - Example: $\sin(30^\circ) = \frac{1}{2}$, $\cos(45^\circ) = \frac{\sqrt{2}}{2}$, $\tan(60^\circ) = \sqrt{3}$ ### Statistics & Probability (Advanced) - **Data Distribution:** How data points are spread. - **Normal Distribution (Bell Curve):** Symmetrical, mean=median=mode. - **Skewness:** Asymmetry in distribution. - **Standard Deviation:** A measure of how spread out numbers are from the mean. - **Correlation:** Relationship between two variables. - **Positive:** As one increases, other increases. - **Negative:** As one increases, other decreases. - **Types of Probability:** - **Theoretical:** Based on mathematical reasoning. - **Experimental:** Based on observed outcomes. - **Compound Events:** - **Independent Events:** Outcome of one doesn't affect the other. $P(A \text{ and } B) = P(A) \times P(B)$ - **Dependent Events:** Outcome of one affects the other. - **Mutually Exclusive Events:** Cannot happen at the same time. $P(A \text{ or } B) = P(A) + P(B)$ ### Analytic Geometry (Coordinate Geometry) - **Coordinate Plane:** $x$-axis (horizontal), $y$-axis (vertical). Points $(x,y)$. - **Distance Formula:** Distance between $(x_1, y_1)$ and $(x_2, y_2)$ is $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$. - **Midpoint Formula:** Midpoint of segment between $(x_1, y_1)$ and $(x_2, y_2)$ is $M = \left(\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2}\right)$. - **Slope of a Line:** $m = \frac{y_2-y_1}{x_2-x_1} = \frac{\text{rise}}{\text{run}}$. - **Equations of Lines:** - **Slope-intercept form:** $y = mx + b$ - **Point-slope form:** $y - y_1 = m(x - x_1)$ - **Standard form:** $Ax + By = C$ - **Conic Sections:** Shapes formed by intersecting a plane with a double cone. - **Circle:** $(x-h)^2 + (y-k)^2 = r^2$ (center $(h,k)$, radius $r$) - **Parabola:** $y = a(x-h)^2 + k$ or $x = a(y-k)^2 + h$ - **Ellipse:** $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ - **Hyperbola:** $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$ or $\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1$ ### Pre-Calculus Essentials - **Advanced Functions:** - **Polynomial Functions:** General form $f(x) = a_n x^n + ... + a_1 x + a_0$. End behavior, roots, multiplicity. - **Rational Functions:** $f(x) = \frac{P(x)}{Q(x)}$. Vertical asymptotes ($Q(x)=0$), horizontal asymptotes. - **Exponential Functions:** $f(x) = a^x$. Growth/decay. - **Logarithmic Functions:** $f(x) = \log_b x$. Inverse of exponential. - **Properties:** $\log_b(MN) = \log_b M + \log_b N$, $\log_b(\frac{M}{N}) = \log_b M - \log_b N$, $\log_b M^p = p \log_b M$, $\log_b b = 1$, $\log_b 1 = 0$. - **Change of Base:** $\log_b x = \frac{\log_c x}{\log_c b}$ - **Trigonometric Functions:** - **Graphs:** Sine, Cosine, Tangent waves (amplitude, period, phase shift, vertical shift). - **Identities:** - Pythagorean: $\sin^2\theta + \cos^2\theta = 1$, $\tan^2\theta + 1 = \sec^2\theta$, $1 + \cot^2\theta = \csc^2\theta$ - Sum/Difference: $\sin(A \pm B) = \sin A \cos B \pm \cos A \sin B$ - Double Angle: $\sin(2\theta) = 2\sin\theta\cos\theta$ - **Solving Trig Equations:** Find angles that satisfy the equation. - **Sequences and Series:** - **Arithmetic:** $a_n = a_1 + (n-1)d$, $S_n = \frac{n}{2}(a_1 + a_n)$ - **Geometric:** $a_n = a_1 r^{n-1}$, $S_n = \frac{a_1(1-r^n)}{1-r}$, $S_\infty = \frac{a_1}{1-r}$ (for $|r| ### Calculus Introduction - **Limits:** The value a function approaches as the input approaches some value. - $\lim_{x \to c} f(x) = L$ - Formal definition: For every $\epsilon > 0$, there exists a $\delta > 0$ such that if $0 ### Discrete Math Basics - **Logic:** - **Propositions:** Statements that are either true or false. - **Connectives:** AND ($\land$), OR ($\lor$), NOT ($\neg$), IMPLIES ($\to$), IFF ($\leftrightarrow$). - **Truth Tables:** Evaluate truth values of compound propositions. - **Quantifiers:** Universal ($\forall$, "for all"), Existential ($\exists$, "there exists"). - **Set Theory:** - **Sets:** Collections of distinct objects. $A = \{1, 2, 3\}$ - **Operations:** - **Union ($\cup$):** Elements in A or B or both. - **Intersection ($\cap$):** Elements in both A and B. - **Complement ($A^c$ or $\bar{A}$):** Elements not in A. - **Subset ($\subseteq$):** All elements of A are in B. - **Cardinality:** Number of elements in a set, $|A|$. - **Combinatorics:** Counting principles. - **Permutations:** Order matters. $P(n, k) = \frac{n!}{(n-k)!}$ - **Combinations:** Order doesn't matter. $C(n, k) = \frac{n!}{k!(n-k)!}$ - **Factorial:** $n! = n \times (n-1) \times ... \times 1$. $0! = 1$. - **Graph Theory:** Study of graphs (vertices and edges). - **Types:** Directed, undirected, weighted. - **Paths, Cycles:** Sequences of connected vertices. ### Physics (Math Applications) - **Mechanics:** - **Displacement, Velocity, Acceleration:** Derivatives of position with respect to time. - $v(t) = \frac{dx}{dt}$, $a(t) = \frac{dv}{dt} = \frac{d^2x}{dt^2}$ - **Force:** Newton's Second Law: $F = ma$. Vectors are crucial for multiple forces. - **Work:** $W = \int F \cdot dx$ (integral of force over distance). - **Kinematics Equations:** For constant acceleration. E.g., $v = v_0 + at$, $x = x_0 + v_0 t + \frac{1}{2}at^2$. - **Waves & Optics:** - **Wave Equation:** Often involves trigonometric functions and partial derivatives. - **Refraction/Reflection:** Geometric principles, Snell's Law ($n_1 \sin\theta_1 = n_2 \sin\theta_2$). - **Thermodynamics:** - **Ideal Gas Law:** $PV = nRT$. - **Calculus for state changes:** Integrals used to calculate work done by gases. - **Electricity & Magnetism:** - **Ohm's Law:** $V = IR$. - **Capacitance, Inductance:** Involve differential equations. - **Maxwell's Equations:** Fundamental equations describing E&M, expressed using vector calculus (divergence, curl). - $\nabla \cdot E = \frac{\rho}{\epsilon_0}$ (Gauss's Law for Electricity) - $\nabla \cdot B = 0$ (Gauss's Law for Magnetism) - $\nabla \times E = -\frac{\partial B}{\partial t}$ (Faraday's Law) - $\nabla \times B = \mu_0 J + \mu_0 \epsilon_0 \frac{\partial E}{\partial t}$ (Ampere-Maxwell Law)