Search Results

Average (Mean)

\text{average} = \frac{\text{sum of values}}{\text{number of values}}

Add all values and divide by how many there are.

Even and Odd

n \bmod 2 = 0 \Rightarrow \text{even}

A number is even if it divides by 2 with no remainder.

Mixed Number to Fraction

a\frac{b}{c} = \frac{ac + b}{c}

Convert a mixed number to an improper fraction.

Reciprocal

\frac{1}{x}

The multiplicative inverse of a number.

Square of a Number

n^2 = n\times n

A number multiplied by itself.

Cube of a Number

n^3 = n\times n\times n

A number multiplied by itself three times.

Divisibility by 3

\text{digit sum divisible by 3}

A number is divisible by 3 if its digit sum is.

Geometry & TrigonometryBasic

Divisibility by 9

\text{digit sum divisible by 9}

A number is divisible by 9 if its digit sum is.

AlgebraIntermediate

Binomial Theorem

(a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k

Expands any power of a binomial into a sum of terms.

AlgebraBasic

Zero Exponent

a^0 = 1

Any nonzero number to the power zero equals one.

AlgebraBasic

Absolute Value

|x| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}

Distance of a number from zero, always non-negative.

AlgebraIntermediate

Geometric Mean

GM = \sqrt[n]{x_1 x_2 \cdots x_n}

The nth root of the product of n numbers.

AlgebraIntermediate

Rational Exponent

a^{m/n} = \sqrt[n]{a^m}

A fractional exponent is a root and a power combined.

AlgebraBasic

Zero Exponent

a^0 = 1, \; a\neq 0

Any nonzero number to the power zero is one.

Perimeter of a Rhombus

P = 4a

All four sides of a rhombus are equal.

Geometry & TrigonometryBasic

Area of a Rhombus

A = \frac{1}{2} d_1 d_2

Area from the two diagonals of a rhombus.

Geometry & TrigonometryIntermediate

Number of Diagonals

D = \frac{n(n-3)}{2}

Count of diagonals in an n-sided polygon.

TrigonometryAdvanced

De Moivre's Theorem

(\cos\theta + i\sin\theta)^n = \cos n\theta + i\sin n\theta

Powers of complex numbers in trig form.

Statistics & ProbabilityIntermediate

Z-Score

z = \frac{x - \mu}{\sigma}

Number of standard deviations a value is from the mean.

Statistics & ProbabilityIntermediate

Combinations

\binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of ways to choose r items from n (order irrelevant).

Statistics & ProbabilityIntermediate

Permutations

P(n,r) = \frac{n!}{(n-r)!}

Number of ordered arrangements of r items from n.

Statistics & ProbabilityAdvanced

Poisson Distribution

P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval.

Statistics & ProbabilityBasic

Factorial

n! = n\times(n-1)\times\cdots\times 2\times 1

Product of all positive integers up to n.

Permutations

P(n,r) = \frac{n!}{(n-r)!}

Number of ordered arrangements of r items from n.

Combinations

C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}

Number of unordered selections of r items from n.

Factorial

n! = n \times (n-1) \times \cdots \times 2 \times 1

Product of all positive integers up to n.

Permutations with Repetition

n^r

Ordered arrangements of r items from n when repetition is allowed.

Permutations of a Multiset

\frac{n!}{n_1! n_2! \cdots n_k!}

Arrangements of n items where some are identical.

Classical Probability

P(A) = \frac{\text{favorable outcomes}}{\text{total outcomes}}

Probability of an event with equally likely outcomes.

Complement Rule

P(A') = 1 - P(A)

Probability that an event does not occur.

Addition Rule

P(A \cup B) = P(A) + P(B) - P(A \cap B)

Probability that A or B occurs.

Multiplication Rule (Independent)

P(A \cap B) = P(A) \cdot P(B)

Probability of both independent events occurring.

Conditional Probability

P(A|B) = \frac{P(A \cap B)}{P(B)}

Probability of A given that B has occurred.

Bayes Theorem

P(A|B) = \frac{P(B|A)\,P(A)}{P(B)}

Reverses conditional probabilities using prior knowledge.

Expected Value

E[X] = \sum_{i} x_i\, p_i

Long-run average value of a random variable.

Variance of a Random Variable

Var(X) = E[X^2] - (E[X])^2

Measures the spread of a random variable around its mean.

Binomial Probability

P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}

Probability of k successes in n independent trials.

Poisson Probability

P(X=k) = \frac{\lambda^k e^{-\lambda}}{k!}

Probability of k events in a fixed interval given average rate λ.

Geometric Probability

P(X=k) = (1-p)^{k-1} p

Probability that the first success occurs on trial k.

Normal Distribution PDF

f(x) = \frac{1}{\sigma\sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Bell-curve probability density function.

Odds

Odds = \frac{P(A)}{1 - P(A)}

Ratio of the probability of an event to its non-occurrence.

Pigeonhole Principle

\lceil n/k \rceil

If n items go into k containers, some container holds at least this many.

Inclusion-Exclusion (3 sets)

|A\cup B\cup C| = |A|+|B|+|C|-|A\cap B|-|A\cap C|-|B\cap C|+|A\cap B\cap C|

Counts the union of overlapping sets without double counting.

Catalan Number

C_n = \frac{1}{n+1}\binom{2n}{n}

Counts many recursive structures such as balanced parentheses.

Data Science & MLIntermediate

Stars and Bars

\binom{n+k-1}{k-1}

Ways to distribute n identical items into k distinct groups.

Sigmoid Function

\sigma(x) = \frac{1}{1 + e^{-x}}

Maps any real number to the range (0, 1); used in logistic regression.

Linear AlgebraAdvanced

Eigenvalue Equation

A\vec{v} = \lambda\vec{v}

Defines eigenvalues and eigenvectors of a matrix.

PhysicsBasic

Wave Speed

v = f\lambda

Speed of a wave from frequency and wavelength.

PhysicsAdvanced

Coulomb's Law

F = k\frac{q_1 q_2}{r^2}

Electrostatic force between two charges.

Physics Laws & TheoremsIntermediate

Coulomb's Law

F = k\frac{q_1 q_2}{r^2}

Electrostatic force between two point charges.

Thermodynamics & HeatIntermediate

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

Combines Boyle, Charles and Gay-Lussac.

Optics & WavesBasic

Wave Speed

v = f\lambda

Speed of a wave from frequency and wavelength.

Optics & WavesAdvanced

Diffraction Grating

d\sin\theta = m\lambda

Condition for constructive interference maxima.

Optics & WavesAdvanced

Double-Slit Fringe Spacing

\Delta y = \frac{\lambda L}{d}

Spacing of interference fringes on a screen.

Optics & WavesIntermediate

Photon Energy (Wavelength)

E = \frac{hc}{\lambda}

Photon energy from its wavelength.

Electricity & MagnetismIntermediate

Coulomb's Law

F = k\frac{q_1 q_2}{r^2}

Force between two point charges.

Fluid MechanicsAdvanced

Reynolds Number

Re = \frac{\rho v L}{\mu}

Predicts laminar vs turbulent flow.

Modern & Quantum PhysicsAdvanced

de Broglie Wavelength

\lambda = \frac{h}{p}

Wavelength associated with a particle.

Modern & Quantum PhysicsIntermediate

Radioactive Decay

N = N_0 e^{-\lambda t}

Number of nuclei remaining over time.

Modern & Quantum PhysicsIntermediate

Half-Life

t_{1/2} = \frac{\ln 2}{\lambda}

Time for half a sample to decay.

Modern & Quantum PhysicsIntermediate

Radioactive Activity

A = \lambda N

Decays per second in a sample.

Modern & Quantum PhysicsAdvanced

Wien's Displacement Law

\lambda_{max} = \frac{b}{T}

Peak emission wavelength of a hot body.

Modern & Quantum PhysicsAdvanced

Compton Shift

\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)

Wavelength change of scattered photons.

Electronics & CircuitsAdvanced

Apparent Power

S = \sqrt{P^2 + Q^2}

Combination of real and reactive power.

Moles from Mass

n = \frac{m}{M}

Number of moles from mass and molar mass.

ChemistryAdvanced

Beer–Lambert Law

A = \varepsilon l c

Absorbance of light by a solution.

ChemistryIntermediate

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

Relates two states of a fixed amount of gas.

Avogadro's Number

N = n \times N_A

Number of particles from moles.

Avogadro Number

N_A = 6.022\times 10^{23}

Number of particles in one mole.

Biology & Life SciencesAdvanced

Number of Particles

N = n\times N_A

Particles from moles and Avogadro number.

ChemistryIntermediate

Combined Gas Law

\frac{P_1 V_1}{T_1} = \frac{P_2 V_2}{T_2}

Combines Boyle, Charles, and Gay-Lussac laws.

Hemocytometer Cell Count

C = \bar{n} \times DF \times 10^4

Cells per mL from a counting-chamber average.

Biology & Life SciencesIntermediate

Bacterial Generations

n = \frac{\log N_t - \log N_0}{\log 2}

Number of divisions over a growth period.

Biology & Life SciencesAdvanced

Recombination Frequency

RF = \frac{\text{recombinants}}{\text{total offspring}} \times 100

Genetic map distance in centimorgans.

Biology & Life SciencesAdvanced

Enzyme Turnover Number

k_{cat} = \frac{V_{max}}{[E]}

Reactions catalyzed per enzyme per second.

Nutrition & FoodAdvanced

Glycemic Load

GL = \frac{GI \times C}{100}

Impact of a food portion on blood sugar, combining quality (GI) and quantity.

Nutrition & FoodBasic

Recipe Scaling Factor

m_{new} = m_{old} \times \frac{n_{want}}{n_{have}}

Rescale each ingredient for a different number of servings.

Nutrition & FoodBasic

Cost per Serving

c_s = \frac{c_{total}}{n}

Total recipe cost divided across the number of servings.

Astronomy & SpaceAdvanced

Redshift

z = \frac{\lambda_{obs} - \lambda_{emit}}{\lambda_{emit}}

Fractional wavelength shift of light.

Astronomy & SpaceIntermediate

Wien's Displacement Law

\lambda_{max} = \frac{b}{T}

Peak emission wavelength of a blackbody.

Astronomy & SpaceAdvanced

Gravitational Redshift

z \approx \frac{GM}{rc^2}

Fractional stretching of light climbing out of gravity.

EngineeringAdvanced

Reynolds Number

Re = \frac{\rho v L}{\mu}

Predicts laminar vs turbulent flow.

Materials & CivilAdvanced

Slenderness Ratio

\lambda = \frac{L_e}{r}

Ratio governing column buckling.

Aviation & AeronauticsBasic

True Airspeed (rule of thumb)

TAS \approx IAS + (IAS \times 0.02 \times \tfrac{alt}{1000})

Quick estimate: true airspeed increases about 2% per 1000 ft of altitude above indicated airspeed.

Aviation & AeronauticsBasic

Rate of Climb

ROC = \dfrac{\Delta h}{t}

Vertical speed is the altitude gained per unit time (ft per minute).

Aviation & AeronauticsIntermediate

Mach Number

M = \dfrac{v}{a}

Ratio of true airspeed to the local speed of sound.

Aviation & AeronauticsAdvanced

Reynolds Number

Re = \dfrac{\rho v L}{\mu}

Ratio of inertial to viscous forces; characterises airflow over an airfoil.

Aviation & AeronauticsAdvanced

Climb Gradient

\gamma = \dfrac{T - D}{W}

Climb angle (as a fraction) equals excess thrust divided by weight.

Aviation & AeronauticsAdvanced

Great Circle Distance

d = R\arccos(\sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos\Delta\lambda)

Shortest distance between two points on the Earth for long-range navigation.

Aviation & AeronauticsIntermediate

Mach Number

M = \frac{v}{a}

Ratio of speed to the local speed of sound.

Aviation & AeronauticsIntermediate

Rate of Climb

ROC = \frac{\text{excess power}}{W}

Vertical speed from excess engine power.

Business & CommerceIntermediate

Break-Even (Units)

Q = \dfrac{Fixed\ Costs}{Price - Variable\ Cost}

The number of units that must be sold to exactly cover all costs.

Everyday MathBasic

Recipe Scaling

Q_{new} = Q_{old} \times \frac{S_{new}}{S_{old}}

Adjust ingredient quantities for a different number of servings.

Everyday MathBasic

Tiles Needed

N = \frac{A_{floor}}{A_{tile}}

Number of tiles to cover a floor.

Time & CalendarAdvanced

Julian Day Number

JDN = \left\lfloor 365.25(Y+4716) \right\rfloor + \left\lfloor 30.6001(M+1) \right\rfloor + D + B - 1524

Continuous day count used by astronomers to compare dates.

Time & CalendarIntermediate

Days Between Two Dates

\Delta = JDN_2 - JDN_1

Number of days separating two calendar dates.

Time & CalendarAdvanced

ISO Week Number

W = \left\lfloor \dfrac{d_o - w_d + 10}{7} \right\rfloor

ISO-8601 week of the year (weeks start on Monday).

Time & CalendarIntermediate

Ordinal Day of Year

N = \left\lfloor \dfrac{275M}{9} \right\rfloor - k\left\lfloor \dfrac{M+9}{12} \right\rfloor + D - 30

Day-of-year number (1–366) for a given date.

Geography & EarthAdvanced

Haversine Distance

d = 2r\arcsin\sqrt{\sin^2\frac{\Delta\phi}{2} + \cos\phi_1\cos\phi_2\sin^2\frac{\Delta\lambda}{2}}

Great-circle distance between two lat/long points.

Geography & EarthBasic

Population Density

D = \frac{P}{A}

Number of people per unit area.

Geography & EarthAdvanced

Radiometric Dating Age

t = \frac{1}{\lambda}\ln\!\left(1 + \frac{D}{P}\right)

Age of a rock from parent and daughter isotopes.

Environmental ScienceIntermediate

Radioactive Decay

N = N_0\left(\frac{1}{2}\right)^{t/t_{1/2}}

Amount remaining after a number of half-lives.

Unit ConversionsBasic

Gigabytes to Megabytes

\text{MB} = \text{GB} \times 1024

Convert gigabytes to megabytes (binary).

Euclidean Algorithm

\gcd(a,b) = \gcd(b, a \bmod b)

Recursively finds the greatest common divisor.

Modular Congruence

a \equiv b \pmod{m}

a and b have the same remainder when divided by m.

Fermat's Little Theorem

a^{p-1} \equiv 1 \pmod{p}

Holds when p is prime and gcd(a,p)=1.

Euler's Totient

\phi(n) = n\prod_{p|n}\left(1 - \frac{1}{p}\right)

Counts integers up to n coprime to n.

Prime Number Theorem

\pi(x) \sim \frac{x}{\ln x}

Approximates the number of primes below x.

LCM from GCD

\text{lcm}(a,b) = \frac{|a b|}{\gcd(a,b)}

Least common multiple via the greatest common divisor.

Sum of First n Integers

\sum_{k=1}^{n} k = \frac{n(n+1)}{2}

Closed form for the sum of the first n natural numbers.

Sum of First n Squares

\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}

Closed form for the sum of the first n squares.

Divisibility Rule for 3

n \text{ divisible by } 3 \iff \text{digit sum divisible by } 3

A number is divisible by 3 if the sum of its digits is.

Divisibility Rule for 9

n \text{ divisible by } 9 \iff \text{digit sum divisible by } 9

A number is divisible by 9 if the sum of its digits is.

Even and Odd Numbers

n = 2k \;(\text{even}), \quad n = 2k+1 \;(\text{odd})

Every integer is either even or odd.

Sum of First n Cubes

\sum_{k=1}^{n} k^3 = \left(\frac{n(n+1)}{2}\right)^2

The sum of the first n cubes equals the square of the sum.

Number of Divisors

d(n) = \prod (a_i + 1)

Count of divisors from prime factorization exponents.

Sum of Divisors

\sigma(n) = \prod \frac{p_i^{a_i+1}-1}{p_i - 1}

Sum of all divisors from prime factorization.

Even Perfect Number

N = 2^{p-1}(2^p - 1)

Euclid-Euler form of even perfect numbers.

Mersenne Number

M_p = 2^p - 1

Numbers one less than a power of two.

Fermat Number

F_n = 2^{2^n} + 1

Numbers of a special exponential form.

Triangular Number

T_n = \frac{n(n+1)}{2}

Sum of the first n natural numbers.

Square Number

S_n = n^2

The nth perfect square.

Pentagonal Number

P_n = \frac{n(3n-1)}{2}

The nth pentagonal figurate number.

Fibonacci Recurrence

F_n = F_{n-1} + F_{n-2}

Each term is the sum of the two before it.

Binet Formula

F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}

Closed form for the nth Fibonacci number.

Golden Ratio

\varphi = \frac{1+\sqrt{5}}{2} \approx 1.618

Ratio where the whole is to the larger as the larger is to the smaller.

Collatz Step

n \to \begin{cases} n/2 & \text{even} \\ 3n+1 & \text{odd} \end{cases}

The rule of the famous unsolved Collatz conjecture.

Wilson Theorem

(p-1)! \equiv -1 \pmod{p}

A primality criterion for p.

Computer ScienceAdvanced

Hamming Distance

d(x,y) = \sum_i [x_i \ne y_i]

Number of positions at which two strings differ.

Bits to Represent n

b = \lceil \log_2(n) \rceil

Minimum number of bits to represent n distinct values.

Binary to Decimal

N = \sum_{i=0}^{k} b_i 2^i

Converts a binary number to decimal.

Decimal to Binary

N = \text{repeated division by } 2

Convert a decimal number to binary by dividing by 2 and reading remainders.

Computer ScienceIntermediate

Hexadecimal to Decimal

N = \sum_{i=0}^{k} h_i 16^i

Convert a hexadecimal number to decimal.

Address Space

N = 2^n

Number of addressable locations for n bits.

Hexadecimal to Decimal

d = \sum h_i \times 16^i

Convert a hex number to decimal by positional weights.

Throughput

X = \frac{N}{T}

Number of operations completed per unit time.

Math Tricks (22)

Nikhilam (Multiply Near Base)

A Vedic sutra for multiplying numbers close to a base (10, 100, 1000, etc.). Works by finding how far each number is from the base.

Urdhva Tiryagbhyam (Vertically & Crosswise)

A general Vedic multiplication method that works for any two numbers. Uses vertical and crosswise multiplication patterns.

Squaring Numbers Near 50

Quick method to square any number near 50 using the identity (50±d)² = 2500 ± 100d + d².

Ekadhikena (Squaring Numbers Ending in 5)

Instantly square any number ending in 5. Multiply the tens digit by (tens digit + 1), then append 25.

Vinculum Numbers

Transform numbers with large digits (6-9) into "vinculum" form using complements from 10, making calculations easier.

Multiply Any 2-Digit Number by 11

Instantly multiply any two-digit number by 11 by splitting and adding the digits.

Divisibility Quick Checks

Quickly check if a number is divisible by 2 through 12 without dividing.

Square Any Number (a±b Method)

Use the identity (a+b)(a-b) + b² = a² to square numbers quickly by going to the nearest easy number.

Multiply by 9, 99, 999 (Nikhilam Special)

Multiplying any number by a string of nines is instant: subtract 1 from the number for the left part, then take the complement for the right part.

Anurupyena (Proportion Multiplication)

When numbers are near a convenient working base that is a multiple of a power of 10, use a proportional base to simplify the multiplication.

Duplex Method for Squaring

The Dwandwa (duplex) method squares any number by summing duplex values digit by digit — a general Vedic squaring technique.

Multiply by 9 the Easy Way

Multiply by 10 and subtract the original number — much faster than multiplying by 9 directly.

Multiply Two Numbers Just Above 100

For numbers a little above 100, use the surplus from 100 to multiply almost instantly.

Percentages with Building Blocks

Any percentage can be built from 10%, 5%, and 1% — quick to find and easy to combine.

Square Numbers Ending in 1

Use (a+1)² = a² + 2a + 1. Square the round number below, add twice it, add 1.

Sum of 1 to N Instantly

Gauss's formula: the sum of the first n whole numbers is n(n+1)/2 — no adding required.

Sum of First N Odd Numbers = N²

The sum of the first n odd numbers is always a perfect square, exactly n².

FOIL for Binomial Products

Multiply two binomials by combining First, Outer, Inner, Last terms.

Product of Numbers Around a Square

Two numbers equally spaced around a middle value m multiply to m² − d², using the difference of squares.

Miles ↔ Kilometres Fast

Multiply miles by 1.6 for km, or divide km by 1.6 for miles. Consecutive Fibonacci numbers are handy approximations.

Estimate a Square Root

Bracket the number between two perfect squares, then refine with a quick average step.

Split a Bill Evenly

Round the total up to a friendly number, divide by the number of people, then adjust down slightly.