Math formulas

Extend the Pythagorean theorem to three coordinates.

Absolute change is a subtraction, independent of the starting scale.

Absolute value is the distance from zero on the number line.

An absolute-value equation splits into two linear cases when c ≥ 0.

Less-than absolute inequalities become a between-bounds interval; greater-than becomes two rays.

Like degrees add (or subtract) their coefficients.

Common difference d; a_n = a1 + (n−1)d; S_n = n(a1+a_n)/2.

Weight each percent by its count, then divide by the total count.

Average rate of change is rise over run on that interval.

Addition is integer addition of the parsed bit strings.

Each bit is a coefficient of a power of 2.

Repeatedly double the fractional part and collect bits.

Parse as integers, subtract, and write the difference in binary.

Left shift multiplies by 2^k; right shift divides by 2^k (floor).

Each bit of the result is 1 only when both inputs have 1 in that place.

A result bit is 1 if either input bit is 1.

A result bit is 1 when the input bits differ.

Ceiling rounds toward +∞.

The unique class modulo mn is constructed from the modular inverses of the moduli.

Sine and cosine (and the other pairs) swap on complementary angles.

Even terms divide by 2; odd terms go to 3n+1 until 1.

Cross-multiply: a d ? b c, with the same sign of the denominators.

Complete the square after factoring a from the quadratic terms.

The conjugate flips the sign of the imaginary part.

Use rectangular arithmetic; division multiplies by the conjugate.

The middle of an odd run is S/n; an even run straddles that mean.

Replace a column of the coefficient matrix by the right-hand side and ratio determinants.

The cross product is perpendicular to both inputs; magnitude is the parallelogram area.

Depress the cubic and solve with the trigonometric or Cardano form.

Sign changes in p(x) bound positive roots; sign changes in p(−x) bound negative roots.

The pair are roots of t² − (sum)t + product = 0.

The digital root is n mod 9, except multiples of 9 (nonzero) become 9.

Direct variation is a constant ratio y/x.

The discriminant decides two real, one real, or complex conjugate roots.

n is divisible by d when the remainder is 0.

u·v = |u||v| cos θ.

Use the standard double-angle identities.

Constant growth doubles in ln(2) divided by ln(1 + r).

exp(x) is the inverse of the natural logarithm.

At each step take the unit fraction 1/⌈b/a⌉ and repeat on the remainder.

Roots of λ² − (tr A)λ + det A = 0.

Enter a positive integer n within the educational size limit. Related: Toy RSA Demonstrator.

Each digit is multiplied by its power of ten.

General real powers use the exponential definition.

Exponential functions scale the base by the coefficient a.

Discrete compound growth multiplies by (1+r) each period.

Enter a and modulus m. If gcd ≠ 1, no modular inverse exists (shown as —). Related: Modular Exponentiation, Toy RSA Demonstrator.

When an inverse exists, a·inv ≡ 1 (mod m).

Find integers whose product is ac and whose sum is b.

Each term is the sum of the previous two.

Floor division rounds the quotient toward −∞.

Floor rounds toward −∞.

FOIL multiplies First, Outer, Inner, Last and collects like terms.

Simplify by dividing by the greatest common divisor.

Addition uses a common denominator.

Apply the exponent to numerator and denominator; negative n takes the reciprocal first.

Enter Integer a and Integer b for the GCD and LCM Calculator; use finite values in the indicated domain. The calculator reports GCD and LCM: GCD is the largest shared divisor; LCM is the least common multiple. When either integer is zero, check the LCM convention required by your application.

The geometric mean is the nth root of the product.

a_n = a1 r^{n−1}; finite sum uses (1−r^n)/(1−r) when r ≠ 1.

Half-angle formulas take square roots of (1±cos θ)/2.

The harmonic mean is n divided by the sum of reciprocals.

The harmonic series of n terms is the sum of reciprocals.

Each hex digit encodes four bits.

Closed ends use brackets; open ends use parentheses.

Inverse variation keeps the product constant.

Sum x_i y_{i+1} − x_{i+1} y_i around the closed loop, then take half the absolute value.

c² = a² + b² − 2ab cos C.

The law of sines relates each side to the sine of the opposite angle.

Compare by cross-multiplication, then list in order.

Compute m, then b = y1 − m x1.

Enter a (coeff of x), b, and c for the Linear Equation Calculator; use finite values in the indicated domain. The calculator reports x: Isolate x by subtracting b and dividing by a. A zero a (coeff of x) means the equation has either no unique solution or infinitely many.

Lerp uses the fractional distance along x.

Definition via change of base.

Change-of-base formula.

Double every second digit from the right; subtract 9 if over 9; total must be divisible by 10.

Manhattan distance sums absolute coordinate differences.

Enter a (1,1), b (1,2), c (2,1), and d (2,2) for the Matrix 2×2 Calculator; use finite values in the indicated domain. The calculator reports Determinant, Inv a, Inv b, Inv c, and Inv d: det = ad−bc; inverse is (1/det) times the adjugate. An inverse exists only when Determinant is nonzero.

The determinant expands along the first row (or any row) by cofactors.

Each entry is a row-column dot product.

tr A = a + d for [[a,b],[c,d]].

Swap off-diagonal entries.

Average the x-coordinates and the y-coordinates.

Enter base a, exponent e, and modulus m (non-zero). Related tools: Extended GCD / Modular Inverse and Toy RSA Demonstrator.

The product of principal square roots is the root of the product (a,b ≥ 0).

The common negative log is minus the base-10 logarithm.

The principal root is exp((1/n) ln x) for x > 0.

Trial division up to √n collects factor pairs.

Each bit is inverted; the integer is interpreted in w bits.

The vertex x-coordinate is −b/(2a).

Clear the denominator and solve the resulting linear system for A and B.

Entries are n choose k.

Divide the part by the whole, then multiply by 100.

Progress is current divided by goal, as a percent.

What is P% of X? Multiply the fraction of 100 by X.

X is what percent of Y? Divide the part by the whole, then scale to 100.

Percent change from A to B measures how much the value grew or shrank relative to the starting amount.

Divide the absolute difference by the absolute theoretical value, then scale by 100.

Percents of percents multiply as fractions of 100.

A percentage-point gap is a simple subtraction of two percent values.

n is a perfect cube when ∛n is an integer.

n is a perfect square when √n is an integer.

Permutations.

Combinations.

Period is 2π/|B|; phase shift is C/B.

Place i from the right is worth 10^i.

y − y1 = m(x − x1) rearranges to y = mx + b.

r is the modulus; θ is atan2(y, x).

Synthetic / long division writes dividend = divisor × quotient + remainder.

A finite set of n elements has 2^n subsets.

Powers of i cycle every four: 1, i, −1, −i.

Enter Integer ≥ 2 for the Prime Factors Calculator; use finite values in the indicated domain. The calculator reports Prime factors and Factor count: Prime factorization writes a number as a product of primes. The input must be a whole number of at least 2.

Enter a, b, and c for the Proportion Calculator; use finite values in the indicated domain. The calculator reports x: Cross-multiply to solve a proportion. a must be nonzero because it is the divisor used to recover x.

Enter a (x²), b (x), and c for the Quadratic Equation Calculator; use finite values in the indicated domain. The calculator reports Discriminant, Root 1, and Root 2: Roots come from the quadratic formula using the discriminant. Set a (x²) nonzero; a negative Discriminant has no real Root 1 or Root 2.

Utilization ρ = λ/μ; mean queue wait λ/(μ(μ−λ)) for ρ < 1.

Any rational root in lowest terms has numerator dividing the constant and denominator dividing the leading coefficient.

The reciprocal is the unique y with xy = 1.

Coprime integers share no prime factor.

Remainder after division.

Pythagoras gives c; the acute angles are arctan of the opposite-over-adjacent ratios.

RMS is the square root of the mean of squares.

Scale, round to nearest integer, then scale back.

Expressions use standard precedence with right-associative powers. Trig and inverse trig respect the DEG/RAD mode; hyperbolic functions are always in radians.

The exponent is the integer floor of log10(|x|).

Scale so the first digit is in a known place, round, then scale back.

Factor out the largest square dividing n.

Two-point slope

x₂ ≠ x₁. Same units on each axis.

Horizontal line

y₂ = y₁ and the points are distinct.

Vertical line

Slope is undefined; do not divide by zero.

Point-slope form

A line through (x₁, y₁) with slope m.

Slope-intercept

b is the y-intercept when the line crosses x = 0.

Slope is rise over run, undefined on a vertical line.

b = y − mx; then y = mx + b at any x.

Percent grade is 100 times rise/run; the angle is arctan of that ratio.

Angle-sum identities expand sine and cosine of a sum or difference.

Use the closed forms already listed on the sequence pages.

Cramer’s rule uses 2×2 determinants when the coefficient determinant is nonzero.

Divide the part duration by the whole duration and scale to 100.

Pick distinct small primes p and q, a public exponent e coprime to φ(n), and a message integer m with 0 ≤ m < n. Related: Modular Exponentiation, Extended GCD / Modular Inverse, Euler Totient.

Never use these keys for real security; production RSA uses large random primes and audited libraries.

T_n = n(n+1)/2. Mode 0 computes T_n; mode 1 solves for n given T.

Union collects unique elements; intersection keeps those in both.

This page takes θ in degrees; cosine and sine use the radian equivalent.

The point on the unit circle at that angle.

Tangent is sine divided by cosine, when cosine is not zero.

Unit rate is a/b with denominator 1.

Divide by the magnitude (must be nonzero).

Multiply the whole by P/100.

Add coordinates independently.

Magnitude is the square root of the sum of squares.

Divide the sum of value×weight by the sum of weights.

From slope m, b = y1 − m x1.