Interactive Reference

Numerical Computing

A library of 541 math & science topics in 12 chapters — each with a formula, variable definitions, an interactive calculator, and two worked examples.

Table of contents · 541 topics · 12 chapters

1 Arithmetic & Number Theory 55 topics

  1. 1.1Addition s = a + b
  2. 1.2Subtraction d = a − b
  3. 1.3Multiplication p = a × b
  4. 1.4Division q = a ÷ b
  5. 1.5Modulo (Remainder) r = a − b·⌊a ÷ b⌋
  6. 1.6Exponentiation y = an
  7. 1.7nth Root y = n√x = x1n
  8. 1.8Square Root y = √x
  9. 1.9Cube Root y = 3√x
  10. 1.10Square y = x2
  11. 1.11Cube y = x3
  12. 1.12Absolute Value y = x
  13. 1.13Sign Function y = sgn(x)
  14. 1.14Reciprocal y = 1x
  15. 1.15Floor y = ⌊x⌋
  16. 1.16Ceiling y = ⌈x⌉
  17. 1.17Round to Decimals y = round(x · 10d)10d
  18. 1.18Truncate y = trunc(x)
  19. 1.19Factorial n! = 1 · 2 · 3 ⋯ n
  20. 1.20Double Factorial n!! = n(n−2)(n−4)⋯
  21. 1.21Greatest Common Divisor g = gcd(a, b)
  22. 1.22Least Common Multiple lcm = a·bgcd(a, b)
  23. 1.23Prime Check n prime ⇔ divisors = {1, n}
  24. 1.24Prime Factorization n = p1a1 · p2a2 ⋯
  25. 1.25nth Prime pn = the n-th prime
  26. 1.26Number of Divisors d(n) = #{ k : k ∣ n }
  27. 1.27Sum of Divisors σ(n) = Σ_{k ∣ n} k
  28. 1.28Perfect Number Check n = Σ proper divisors
  29. 1.29Fibonacci Number Fn = Fn−1 + Fn−2
  30. 1.30Lucas Number Ln = Ln−1 + Ln−2
  31. 1.31Triangular Number Tn = n(n+1)2
  32. 1.32Catalan Number Cn = C(2n, n)n+1
  33. 1.33Binomial Coefficient C(n,k) = n!k!(n−k)!
  34. 1.34Permutations (nPr) P(n,r) = n!n−r!
  35. 1.35Combinations (nCr) C(n,r) = n!r!(n−r)!
  36. 1.36Percentage Of y = (p100) · x
  37. 1.37Percentage Change Δ% = xn − x0x0 · 100
  38. 1.38Percent of Total p = partwhole · 100
  39. 1.39Sum of Digits S = Σ di
  40. 1.40Digit Count len(n)
  41. 1.41Reverse Digits reverse(n)
  42. 1.42Base Converter n10 → n_b
  43. 1.43Natural Logarithm y = ln(x)
  44. 1.44Logarithm Base 10 y = log10(x)
  45. 1.45Logarithm Base 2 y = log2(x)
  46. 1.46Logarithm Base n log_b(x) = ln xln b
  47. 1.47Exponential eˣ y = ex
  48. 1.48Collatz Steps n → n2 (even) or 3n+1 (odd)
  49. 1.49Apply Percentage Increase y = x · (1 + p100)
  50. 1.50Apply Percentage Decrease y = x · (1 − p100)
  51. 1.51GCD of a List gcd(a1, a2, …, an)
  52. 1.52LCM of a List lcm(a1, a2, …, an)
  53. 1.53Even or Odd n mod 2
  54. 1.54Round to Nearest Multiple y = m · round(xm)
  55. 1.55Modular Exponentiation r = be mod m

2 Algebra & Equations 47 topics

  1. 2.1Linear Equation Solver x = −ba
  2. 2.2Line From Two Points y = m·x + c
  3. 2.3Quadratic Equation Solver x = −b ± √b2−4ac2a
  4. 2.4Quadratic Discriminant Δ = b2 − 4ac
  5. 2.5Quadratic Vertex (h, k) = (−b2a, c − b24a)
  6. 2.6Sum of Quadratic Roots x1 + x2 = −ba
  7. 2.7Product of Quadratic Roots x1 · x2 = ca
  8. 2.8Cubic Real Root ax3 + bx2 + cx + d = 0
  9. 2.92×2 Linear System x = DxD, y = D_yD
  10. 2.10Complex Addition (a+bi) + (c+di) = (a+c)+(b+d)i
  11. 2.11Complex Subtraction (a+bi) − (c+di)
  12. 2.12Complex Multiplication (ac−bd) + (ad+bc)i
  13. 2.13Complex Division a+bic+di
  14. 2.14Complex Modulus a+bi = √a2+b2
  15. 2.15Complex Argument arg = atan2(b, a)
  16. 2.16Complex Conjugate conj(a+bi) = a − bi
  17. 2.17Complex Power (De Moivre) rn(cos nθ + i sin nθ)
  18. 2.18Polar → Rectangular Complex z = r·cosθ + i·r·sinθ
  19. 2.19Polynomial Evaluation p(x) = Σ ck xk
  20. 2.20Polynomial Derivative p'(x) coefficients
  21. 2.21Remainder Theorem remainder = p(r)
  22. 2.22Arithmetic Sequence Term an = a1 + (n−1)d
  23. 2.23Arithmetic Series Sum Sn = n2 · (2a1 + (n−1)d)
  24. 2.24Geometric Sequence Term an = a1 · rn−1
  25. 2.25Geometric Series Sum Sn = a11 − rn1 − r
  26. 2.26Infinite Geometric Sum S = a11 − r
  27. 2.27Sum of First n Integers Σ = n(n+1)2
  28. 2.28Sum of First n Squares Σ = n(n+1)2n+16
  29. 2.29Sum of First n Cubes Σ = (n(n+1)2)2
  30. 2.30Harmonic Partial Sum Hn = Σ 1k
  31. 2.31Exponential Growth N = N0 · ert
  32. 2.32Exponential Decay N = N0 · e−rt
  33. 2.33Doubling Time t = ln 2r
  34. 2.34Half-Life Remaining N = N0 · (12)tT
  35. 2.35Solve aˣ = b x = ln bln a
  36. 2.36Absolute Difference a − b
  37. 2.37Proportion Solver x = b·ca
  38. 2.38Direct Variation y = k·x
  39. 2.39Inverse Variation y = kx
  40. 2.40Linear Interpolation (2 points) y = y1 + (x−x1)y2−y1x2−x1
  41. 2.41Percent Error exp − truetrue · 100
  42. 2.42Weighted Average (2 values) w1x1 + w2x2w1 + w2
  43. 2.43Scientific Notation → Decimal x = m · 10e
  44. 2.44Geometric Mean (2 values) √a·b
  45. 2.45Completing the Square h = −b2a, k = c − b24a
  46. 2.46Integer Division a = b·q + r
  47. 2.47Fraction to Decimal d = ab

3 Linear Algebra 29 topics

  1. 3.1Vector Addition w = u + v, wi = ui + vi
  2. 3.2Vector Subtraction w = u − v
  3. 3.3Scalar × Vector w = k·v
  4. 3.4Dot Product u · v = Σ ui vi
  5. 3.5Cross Product (3D) u × v
  6. 3.6Vector Magnitude v = √Σ vi2
  7. 3.7Vector Normalization v̂ = vv
  8. 3.8Angle Between Vectors cos θ = u·vuv
  9. 3.9Vector Projection proj_v u = (u·vv·v) v
  10. 3.10Vector Distance u − v
  11. 3.11Scalar Triple Product u · (v × w)
  12. 3.12Matrix Addition C = A + B
  13. 3.13Matrix Subtraction C = A − B
  14. 3.14Scalar × Matrix C = k·A
  15. 3.15Matrix Multiplication Cij = Σk Aik Bkj
  16. 3.16Matrix × Vector y = A·x
  17. 3.17Matrix Transpose (AT)ij = Aji
  18. 3.18Matrix Determinant det(A)
  19. 3.19Matrix Trace tr(A) = Σ Aii
  20. 3.20Matrix Inverse A·A−1 = I
  21. 3.21Identity Matrix Iij = 1 if i=j else 0
  22. 3.22Linear System Solver x = A−1 b
  23. 3.23Matrix Power An
  24. 3.242×2 Eigenvalues λ = tr ± √tr2−4det2
  25. 3.25Matrix Rank rank(A)
  26. 3.26Outer Product (u ⊗ v)ij = ui vj
  27. 3.27Hadamard Product (A ∘ B)ij = Aij Bij
  28. 3.28Frobenius Norm A_F = √Σ Aij2
  29. 3.29Symmetric Matrix Check A = AT ?

4 Calculus & Numerical Methods 33 topics

  1. 4.1Numerical Derivative f'(x) ≈ f(x+h) − f(x−h)2h
  2. 4.2Second Derivative f''(x) ≈ f(x+h) − 2f(x) + f(x−h)h2
  3. 4.3Partial Derivative ∂f/∂x ∂f∂x ≈ f(x+h,y) − f(x−h,y)2h
  4. 4.4Partial Derivative ∂f/∂y ∂f∂y ≈ f(x,y+h) − f(x,y−h)2h
  5. 4.5Gradient Magnitude (2D) ∇f = √(∂f∂x)2 + (∂f∂y)2
  6. 4.6Trapezoidal Integration ∫ab f dx ≈ h(½f0 + f1 + … + ½fn)
  7. 4.7Simpson's Rule Integration ∫ab f dx ≈ (h3)(f0 + 4f1 + 2f2 + … + fn)
  8. 4.8Midpoint Rule Integration ∫ab f dx ≈ h Σ f(midpoints)
  9. 4.9Left Riemann Sum ∫ab f dx ≈ h Σ f(left points)
  10. 4.10Right Riemann Sum ∫ab f dx ≈ h Σ f(right points)
  11. 4.11Double Integral (rectangle) ∬ f dA
  12. 4.12Newton-Raphson Root xn+1 = xn − f(xn)f'(xn)
  13. 4.13Bisection Root repeatedly halve [a, b]
  14. 4.14Secant Method Root xn+1 = xn − fnxn−xn−1fn−fn−1
  15. 4.15False Position Root c = b − f(b)b−af(b)−f(a)
  16. 4.16Fixed-Point Iteration xn+1 = g(xn)
  17. 4.17Euler's Method (ODE) yn+1 = yn + h·f(xn, yn)
  18. 4.18Runge-Kutta 2 (ODE) yn+1 = yn + h·f(x+h2, y+h2·k1)
  19. 4.19Runge-Kutta 4 (ODE) yn+1 = yn + h6(k1+2k2+2k3+k4)
  20. 4.20Linear Interpolation y = y0 + (x−x0)y1−y0x1−x0
  21. 4.21Lagrange Interpolation p(x) = Σ yj Lj(x)
  22. 4.22Taylor Series of eˣ ex ≈ Σ_{k=0}{n−1} xkk!
  23. 4.23Taylor Series of sin x sin x ≈ Σ (−1)k x2k+12k+1!
  24. 4.24Taylor Series of cos x cos x ≈ Σ (−1)k x2k2k!
  25. 4.25Maclaurin Series ln(1+x) ln(1+x) ≈ Σ (−1)k+1 xkk
  26. 4.26Arc Length (numerical) L = ∫ab√1 + f'(x)2 dx
  27. 4.27Numerical Limit lim_{x→a} f(x)
  28. 4.28Average Value of a Function f̄ = (1b−a) ∫ab f dx
  29. 4.29RMS of a Function √(1b−a) ∫ab f2 dx
  30. 4.30Area Between Curves ∫ab f(x) − g(x) dx
  31. 4.31Curvature of y=f(x) κ = f''1 + f'2
  32. 4.32Newton's Law of Cooling T = T_a + (T0 − T_a) e−kt
  33. 4.33Logistic Growth Value P = K1 + (K−P0P0) e−rt

5 Statistics & Probability 52 topics

  1. 5.1Mean (Average) x̄ = Σ xin
  2. 5.2Median middle of sorted values
  3. 5.3Mode argmax frequency
  4. 5.4Range R = max − min
  5. 5.5Midrange max + min2
  6. 5.6Population Variance σ2 = Σ (xi − μ)2N
  7. 5.7Sample Variance s2 = Σ (xi − x̄)2n−1
  8. 5.8Population Std Deviation σ = √σ2
  9. 5.9Sample Std Deviation s = √s2
  10. 5.10Sum Σ xi
  11. 5.11Product Π xi
  12. 5.12Count n = data
  13. 5.13Minimum min(data)
  14. 5.14Maximum max(data)
  15. 5.15Geometric Mean G = (Π xi)1n
  16. 5.16Harmonic Mean H = nΣ 1xi
  17. 5.17Root Mean Square RMS = √Σ xi2n
  18. 5.18Weighted Mean x̄_w = Σ wi xiΣ wi
  19. 5.19Z-Score z = x − μσ
  20. 5.20Coefficient of Variation CV = sx̄ · 100
  21. 5.21Percentile Pp via rank interpolation
  22. 5.22First Quartile (Q1) Q1 = P25
  23. 5.23Third Quartile (Q3) Q3 = P75
  24. 5.24Interquartile Range IQR = Q3 − Q1
  25. 5.25Mean Absolute Deviation MAD = Σ xi − x̄n
  26. 5.26Skewness g1 = (1n) Σ (xi−μσ)3
  27. 5.27Excess Kurtosis g2 = (1n) Σ (xi−μσ)4 − 3
  28. 5.28Covariance (sample) cov = Σ (xi−x̄)(yi−ȳ)n−1
  29. 5.29Pearson Correlation r = cov(x,y)sx s_y
  30. 5.30Regression Slope m = nΣxy − ΣxΣynΣx2 − (Σx)2
  31. 5.31Linear Regression (full) y = m·x + b
  32. 5.32Coefficient of Determination R2 = r2
  33. 5.33Standard Error of the Mean SE = s√n
  34. 5.3495% Confidence Interval x̄ ± 1.96 · s√n
  35. 5.35Normal PDF f(x) = e−x−μ2σ2σ√2π
  36. 5.36Normal CDF Φ(x−μσ)
  37. 5.37Binomial Probability C(n,k) pk (1−p)n−k
  38. 5.38Binomial Mean μ = n·p
  39. 5.39Binomial Variance σ2 = n·p·(1−p)
  40. 5.40Poisson Probability P = λk e−λk!
  41. 5.41Geometric Distribution P = (1−p)k−1 · p
  42. 5.42Exponential PDF f(x) = λ e−λx
  43. 5.43Exponential CDF F(x) = 1 − e−λx
  44. 5.44Uniform PDF f = 1b − a on [a, b]
  45. 5.45Basic Probability P = favorabletotal
  46. 5.46Odds → Probability P = aa + b
  47. 5.47Probability → Odds odds = p : (1 − p)
  48. 5.48Expected Value E[X] = Σ xi pi
  49. 5.49Standard Score → Percentile percentile = Φ(z) · 100
  50. 5.50Five-Number Summary {min, Q1, median, Q3, max}
  51. 5.51Z-Score from Percentile z = Φ−1(p)
  52. 5.52Standard Error (from s, n) SE = s√n

6 Geometry 63 topics

  1. 6.1Circle Area A = π r2
  2. 6.2Semicircle Area A = ½ π r2
  3. 6.3Triangle Area (base × height) A = ½ b h
  4. 6.4Triangle Area (Heron) A = √s(s−a)(s−b)(s−c), s = a+b+c2
  5. 6.5Equilateral Triangle Area A = (√34) a2
  6. 6.6Rectangle Area A = l · w
  7. 6.7Square Area A = a2
  8. 6.8Parallelogram Area A = b · h
  9. 6.9Rhombus Area A = ½ d1 d2
  10. 6.10Trapezoid Area A = ½ (a + b) h
  11. 6.11Ellipse Area A = π a b
  12. 6.12Regular Polygon Area A = n s24 tan(πn)
  13. 6.13Circular Sector Area A = ½ r2 θ_rad
  14. 6.14Annulus Area A = π (R2 − r2)
  15. 6.15Circle Circumference C = 2 π r
  16. 6.16Rectangle Perimeter P = 2(l + w)
  17. 6.17Square Perimeter P = 4 a
  18. 6.18Triangle Perimeter P = a + b + c
  19. 6.19Regular Polygon Perimeter P = n · s
  20. 6.20Circular Arc Length L = r · θ_rad
  21. 6.21Ellipse Perimeter (approx) P ≈ π[3(a+b) − √(3a+b)(a+3b)]
  22. 6.22Circle Diameter d = 2 r
  23. 6.23Circle Radius from Area r = √Aπ
  24. 6.24Circle Radius from Circumference r = C2π
  25. 6.25Sphere Volume V = 43 π r3
  26. 6.26Hemisphere Volume V = 23 π r3
  27. 6.27Cube Volume V = a3
  28. 6.28Cuboid Volume V = l · w · h
  29. 6.29Cylinder Volume V = π r2 h
  30. 6.30Cone Volume V = 13 π r2 h
  31. 6.31Pyramid Volume V = 13 B h
  32. 6.32Prism Volume V = B · h
  33. 6.33Regular Tetrahedron Volume V = a36√2
  34. 6.34Torus Volume V = 2 π2 R r2
  35. 6.35Ellipsoid Volume V = 43 π a b c
  36. 6.36Spherical Cap Volume V = π h2 3r − h3
  37. 6.37Sphere Surface Area A = 4 π r2
  38. 6.38Cube Surface Area A = 6 a2
  39. 6.39Cuboid Surface Area A = 2(lw + lh + wh)
  40. 6.40Cylinder Surface Area A = 2π r (r + h)
  41. 6.41Cone Surface Area A = π r (r + √r2+h2)
  42. 6.42Hemisphere Surface Area A = 3 π r2
  43. 6.43Cylinder Lateral Area A = 2 π r h
  44. 6.44Cone Lateral Area A = π r √r2+h2
  45. 6.45Distance Between 2 Points d = √(x2−x1)2 + (y2−y1)2
  46. 6.46Distance in 3D d = √Δx2 + Δy2 + Δz2
  47. 6.47Midpoint (2D) M = (x1+x22, y1+y22)
  48. 6.48Slope of a Line m = y2 − y1x2 − x1
  49. 6.49Pythagorean Theorem c = √a2 + b2
  50. 6.50Pythagorean Leg b = √c2 − a2
  51. 6.51Point to Line Distance d = a x0 + b y0 + c√a2+b2
  52. 6.52Triangle Area (coordinates) A = ½x1(y2−y3)+x2(y3−y1)+x3(y1−y2)
  53. 6.53Triangle Centroid G = (Σxi3, Σyi3)
  54. 6.54Polygon Interior Angle θ = (n−2)·180n
  55. 6.55Regular Polygon Apothem a = s2 tan(πn)
  56. 6.56Rectangle Diagonal d = √l2 + w2
  57. 6.57Cuboid Space Diagonal d = √l2 + w2 + h2
  58. 6.58Regular Polygon Circumradius R = s2 sin(πn)
  59. 6.59Kite Area A = ½ d1 d2
  60. 6.60Cone Slant Height ℓ = √r2 + h2
  61. 6.61Conical Frustum Volume V = ⅓ π h (R2 + R r + r2)
  62. 6.62Polygon Area (apothem) A = ½ · perimeter · apothem
  63. 6.63Great-Circle Distance d = 2R·asin(√a)

7 Trigonometry 33 topics

  1. 7.1Sine y = sin θ
  2. 7.2Cosine y = cos θ
  3. 7.3Tangent y = tan θ
  4. 7.4Cosecant csc θ = 1sin θ
  5. 7.5Secant sec θ = 1cos θ
  6. 7.6Cotangent cot θ = 1tan θ
  7. 7.7Arcsine θ = asin(x)
  8. 7.8Arccosine θ = acos(x)
  9. 7.9Arctangent θ = atan(x)
  10. 7.10Atan2 θ = atan2(y, x)
  11. 7.11Hyperbolic Sine sinh x = ex − e−x2
  12. 7.12Hyperbolic Cosine cosh x = ex + e−x2
  13. 7.13Hyperbolic Tangent tanh x = sinh xcosh x
  14. 7.14Degrees → Radians rad = deg · π180
  15. 7.15Radians → Degrees deg = rad · 180π
  16. 7.16Degrees → Gradians grad = deg · 109
  17. 7.17Gradians → Degrees deg = grad · 910
  18. 7.18Law of Cosines (side) c = √a2 + b2 − 2ab cos C
  19. 7.19Law of Cosines (angle) C = acos(a2+b2−c22ab)
  20. 7.20Law of Sines (side) b = a · sin Bsin A
  21. 7.21Right Triangle Opposite Side opp = h · sin θ
  22. 7.22Right Triangle Adjacent Side adj = h · cos θ
  23. 7.23Hypotenuse from Opposite & Angle h = oppsin θ
  24. 7.24Angle from Opposite & Adjacent θ = atan(oppadj)
  25. 7.25Chord Length L = 2 r sin(θ2)
  26. 7.26Sine Double Angle sin 2θ = 2 sin θ cos θ
  27. 7.27Cosine Double Angle cos 2θ = cos2θ − sin2θ
  28. 7.28Polar → Cartesian x = r cos θ, y = r sin θ
  29. 7.29Cartesian → Polar r = √x2+y2, θ = atan2(y, x)
  30. 7.30Sector Area (angle) A = (θ360) π r2
  31. 7.31Inverse Hyperbolic Sine asinh(x) = ln(x + √x2+1)
  32. 7.32Inverse Hyperbolic Cosine acosh(x) = ln(x + √x2−1)
  33. 7.33Inverse Hyperbolic Tangent atanh(x) = ½ ln(1+x1−x)

8 Finance 41 topics

  1. 8.1Simple Interest I = P · r · t
  2. 8.2Simple Interest Total A = P(1 + r·t)
  3. 8.3Compound Interest (amount) A = P(1 + rn)n t
  4. 8.4Compound Interest Earned I = P[(1 + rn)n t − 1]
  5. 8.5Continuous Compounding A = P · er t
  6. 8.6Future Value FV = PV(1 + r)n
  7. 8.7Present Value PV = FV1 + r
  8. 8.8Future Value of Annuity FV = PMT · (1 + r)n − 1r
  9. 8.9Present Value of Annuity PV = PMT · 1 − (1 + r)−nr
  10. 8.10Loan Payment (Amortized) M = P·r1 − (1 + r)−n
  11. 8.11Loan Total Interest Total = M·n − P
  12. 8.12CAGR CAGR = (V_eV_b)1t − 1
  13. 8.13Return on Investment ROI = gain − costcost · 100
  14. 8.14Net Present Value NPV = Σ CFt1 + r
  15. 8.15Internal Rate of Return 0 = Σ CFt1 + IRR
  16. 8.16Payback Period T = costannual cash flow
  17. 8.17Straight-Line Depreciation D = cost − salvagelife
  18. 8.18Declining Balance Depreciation D1 = cost · (2life)
  19. 8.19Markup Price price = cost · (1 + markup%)
  20. 8.20Profit Margin margin = rev − costrev · 100
  21. 8.21Gross Profit GP = revenue − COGS
  22. 8.22Discounted Price price · (1 − discount%)
  23. 8.23Discount Percent orig − saleorig · 100
  24. 8.24Break-Even Units Q = FCprice − variable
  25. 8.25Break-Even Revenue Rev = FCcontribution margin ratio
  26. 8.26Contribution Margin CM = price − variable cost
  27. 8.27Effective Annual Rate EAR = (1 + rn)n − 1
  28. 8.28Rule of 72 t ≈ 72rate%
  29. 8.29Real Interest Rate r_real = 1+r1+i − 1
  30. 8.30Inflation-Adjusted Value Real = nominal1 + i
  31. 8.31Dividend Yield yield = dividendprice · 100
  32. 8.32Price/Earnings Ratio PE = priceEPS
  33. 8.33Sales Tax Total total = price · (1 + tax%)
  34. 8.34Tip Calculator tip = bill · tip%100
  35. 8.35Extract VAT from Gross VAT = gross · rate100 + rate
  36. 8.36Monthly Savings for Goal PMT = FV·r(1+r)n − 1
  37. 8.37Down Payment Amount DP = price · percent100
  38. 8.38Debt-to-Income Ratio DTI = debtincome · 100
  39. 8.39Annual Percentage Yield APY = (1 + rn)n − 1
  40. 8.40Future Value (Annuity Due) FV = PMT·(1+r)n−1r·(1+r)
  41. 8.41Solve Compound Rate r = (FVPV)1n − 1

9 Physics 68 topics

  1. 9.1Average Velocity v = dt
  2. 9.2Acceleration a = v − ut
  3. 9.3Displacement (uniform accel) s = u·t + ½ a t2
  4. 9.4Final Velocity (v = u + at) v = u + a·t
  5. 9.5Velocity² (v² = u² + 2as) v = √u2 + 2 a s
  6. 9.6Projectile Range R = v2 sin(2θ)g
  7. 9.7Projectile Max Height H = v2 sin2θ2g
  8. 9.8Projectile Flight Time T = 2 v sinθg
  9. 9.9Free Fall Distance d = ½ g t2
  10. 9.10Free Fall Velocity v = g · t
  11. 9.11Newton's Second Law F = m · a
  12. 9.12Weight W = m · g
  13. 9.13Momentum p = m · v
  14. 9.14Impulse J = F · Δt
  15. 9.15Friction Force f = μ · N
  16. 9.16Centripetal Acceleration a = v2r
  17. 9.17Centripetal Force F = m v2r
  18. 9.18Work Done W = F · d · cos θ
  19. 9.19Power (work/time) P = Wt
  20. 9.20Power (force × velocity) P = F · v
  21. 9.21Kinetic Energy KE = ½ m v2
  22. 9.22Gravitational PE PE = m · g · h
  23. 9.23Elastic Potential Energy PE = ½ k x2
  24. 9.24Hooke's Law F = k · x
  25. 9.25Newton's Gravitation F = G m1 m2r2
  26. 9.26Gravitational Field Strength g = G Mr2
  27. 9.27Escape Velocity v = √2 G Mr
  28. 9.28Orbital Velocity v = √G Mr
  29. 9.29Wave Speed v = f · λ
  30. 9.30Frequency from Period f = 1T
  31. 9.31Period from Frequency T = 1f
  32. 9.32Wavelength λ = vf
  33. 9.33Simple Pendulum Period T = 2π √Lg
  34. 9.34Pressure P = FA
  35. 9.35Hydrostatic Pressure P = ρ g h
  36. 9.36Density ρ = mV
  37. 9.37Buoyant Force F = ρ V g
  38. 9.38Volumetric Flow Rate Q = A · v
  39. 9.39Heat Energy (Q = mcΔT) Q = m c ΔT
  40. 9.40Linear Thermal Expansion ΔL = α L0 ΔT
  41. 9.41Ideal Gas Pressure P = n R TV
  42. 9.42Ideal Gas Volume V = n R TP
  43. 9.43Carnot Efficiency η = 1 − T_cT_h
  44. 9.44Ohm's Law (voltage) V = I · R
  45. 9.45Ohm's Law (current) I = VR
  46. 9.46Ohm's Law (resistance) R = VI
  47. 9.47Electrical Power (P = VI) P = V · I
  48. 9.48Power (P = I²R) P = I2 · R
  49. 9.49Resistors in Series R = R1 + R2 + …
  50. 9.50Resistors in Parallel 1R = Σ 1Ri
  51. 9.51Capacitors in Parallel C = C1 + C2 + …
  52. 9.52Capacitors in Series 1C = Σ 1Ci
  53. 9.53Coulomb's Law F = k q1 q2r2
  54. 9.54Electric Field (point charge) E = k qr2
  55. 9.55Capacitor Energy E = ½ C V2
  56. 9.56Electric Charge Q = I · t
  57. 9.57Snell's Law (refraction angle) θ2 = asin(n1 sinθ1n2)
  58. 9.58Thin Lens (image distance) 1f = 1do + 1di
  59. 9.59Magnification M = −dido
  60. 9.60Photon Energy E = h · f
  61. 9.61Mass–Energy Equivalence E = m c2
  62. 9.62Rotational Kinetic Energy KE = ½ I ω2
  63. 9.63Angular Velocity ω = 2π f
  64. 9.64Torque τ = r · F · sin θ
  65. 9.65De Broglie Wavelength λ = hm v
  66. 9.66Efficiency η = outputinput · 100
  67. 9.67Specific Heat Capacity c = Qm ΔT
  68. 9.68Time from Velocity Change t = v − ua

10 Chemistry 22 topics

  1. 10.1Moles from Mass n = mM
  2. 10.2Mass from Moles m = n · M
  3. 10.3Molarity C = nV
  4. 10.4Moles from Molarity n = C · V
  5. 10.5Solution Volume V = nC
  6. 10.6Dilution (final conc.) C2 = C1 V1V2
  7. 10.7Dilution (final volume) V2 = C1 V1C2
  8. 10.8Ideal Gas Moles n = P VR T
  9. 10.9Boyle's Law V2 = P1 V1P2
  10. 10.10Charles's Law V2 = V1 T2T1
  11. 10.11pH from [H⁺] pH = −log10[H+]
  12. 10.12[H⁺] from pH [H+] = 10−pH
  13. 10.13pOH from [OH⁻] pOH = −log10[OH−]
  14. 10.14pH from pOH pH = 14 − pOH
  15. 10.15Percent Yield % = actualtheoretical · 100
  16. 10.16Percent Composition % = elem masstotal mass · 100
  17. 10.17Mole Fraction x = n_An_A + n_B
  18. 10.18Mass Percent (solution) % = solutesolution · 100
  19. 10.19Parts Per Million ppm = solutesolution · 106
  20. 10.20Average Atomic Mass Σ (massi · abundancei)
  21. 10.21First-Order Half-Life t½ = ln 2k
  22. 10.22Gas Volume at STP V = n · 22.414 L

11 Unit Conversions 90 topics

  1. 11.1Meters → Kilometers km = m × 0.001
  2. 11.2Meters → Centimeters cm = m × 100
  3. 11.3Meters → Miles mi = m × 0.00062137119
  4. 11.4Meters → Feet ft = m × 3.2808399
  5. 11.5Meters → Inches in = m × 39.370079
  6. 11.6Kilometers → Meters m = km × 1000
  7. 11.7Kilometers → Centimeters cm = km × 100000
  8. 11.8Kilometers → Miles mi = km × 0.62137119
  9. 11.9Kilometers → Feet ft = km × 3280.8399
  10. 11.10Kilometers → Inches in = km × 39370.079
  11. 11.11Centimeters → Meters m = cm × 0.01
  12. 11.12Centimeters → Kilometers km = cm × 0.00001
  13. 11.13Centimeters → Miles mi = cm × 0.0000062137119
  14. 11.14Centimeters → Feet ft = cm × 0.032808399
  15. 11.15Centimeters → Inches in = cm × 0.39370079
  16. 11.16Miles → Meters m = mi × 1609.344
  17. 11.17Miles → Kilometers km = mi × 1.609344
  18. 11.18Miles → Centimeters cm = mi × 160934.4
  19. 11.19Miles → Feet ft = mi × 5280
  20. 11.20Miles → Inches in = mi × 63360
  21. 11.21Feet → Meters m = ft × 0.3048
  22. 11.22Feet → Kilometers km = ft × 0.0003048
  23. 11.23Feet → Centimeters cm = ft × 30.48
  24. 11.24Feet → Miles mi = ft × 0.00018939394
  25. 11.25Feet → Inches in = ft × 12
  26. 11.26Inches → Meters m = in × 0.0254
  27. 11.27Inches → Kilometers km = in × 0.0000254
  28. 11.28Inches → Centimeters cm = in × 2.54
  29. 11.29Inches → Miles mi = in × 0.000015782828
  30. 11.30Inches → Feet ft = in × 0.083333333
  31. 11.31Kilograms → Grams g = kg × 1000
  32. 11.32Kilograms → Pounds lb = kg × 2.2046226
  33. 11.33Kilograms → Ounces oz = kg × 35.273962
  34. 11.34Grams → Kilograms kg = g × 0.001
  35. 11.35Grams → Pounds lb = g × 0.0022046226
  36. 11.36Grams → Ounces oz = g × 0.035273962
  37. 11.37Pounds → Kilograms kg = lb × 0.45359237
  38. 11.38Pounds → Grams g = lb × 453.59237
  39. 11.39Pounds → Ounces oz = lb × 16
  40. 11.40Ounces → Kilograms kg = oz × 0.028349523
  41. 11.41Ounces → Grams g = oz × 28.349523
  42. 11.42Ounces → Pounds lb = oz × 0.0625
  43. 11.43Liters → Milliliters mL = L × 1000
  44. 11.44Liters → US Gallons gal = L × 0.26417205
  45. 11.45Liters → US Fluid Ounces floz = L × 33.814023
  46. 11.46Milliliters → Liters L = mL × 0.001
  47. 11.47Milliliters → US Gallons gal = mL × 0.00026417205
  48. 11.48Milliliters → US Fluid Ounces floz = mL × 0.033814023
  49. 11.49US Gallons → Liters L = gal × 3.7854118
  50. 11.50US Gallons → Milliliters mL = gal × 3785.4118
  51. 11.51US Gallons → US Fluid Ounces floz = gal × 128
  52. 11.52US Fluid Ounces → Liters L = floz × 0.02957353
  53. 11.53US Fluid Ounces → Milliliters mL = floz × 29.57353
  54. 11.54US Fluid Ounces → US Gallons gal = floz × 0.0078125
  55. 11.55Meters per second → Kilometers per hour kmh = mps × 3.6
  56. 11.56Meters per second → Miles per hour mph = mps × 2.2369363
  57. 11.57Kilometers per hour → Meters per second mps = kmh × 0.27777778
  58. 11.58Kilometers per hour → Miles per hour mph = kmh × 0.62137119
  59. 11.59Miles per hour → Meters per second mps = mph × 0.44704
  60. 11.60Miles per hour → Kilometers per hour kmh = mph × 1.609344
  61. 11.61Seconds → Minutes min = s × 0.016666667
  62. 11.62Seconds → Hours h = s × 0.00027777778
  63. 11.63Seconds → Days day = s × 0.000011574074
  64. 11.64Minutes → Seconds s = min × 60
  65. 11.65Minutes → Hours h = min × 0.016666667
  66. 11.66Minutes → Days day = min × 0.00069444444
  67. 11.67Hours → Seconds s = h × 3600
  68. 11.68Hours → Minutes min = h × 60
  69. 11.69Hours → Days day = h × 0.041666667
  70. 11.70Days → Seconds s = day × 86400
  71. 11.71Days → Minutes min = day × 1440
  72. 11.72Days → Hours h = day × 24
  73. 11.73Bytes → Kilobytes KB = B × 0.0009765625
  74. 11.74Bytes → Megabytes MB = B × 9.5367432e-7
  75. 11.75Bytes → Gigabytes GB = B × 9.3132257e-10
  76. 11.76Kilobytes → Bytes B = KB × 1024
  77. 11.77Kilobytes → Megabytes MB = KB × 0.0009765625
  78. 11.78Kilobytes → Gigabytes GB = KB × 9.5367432e-7
  79. 11.79Megabytes → Bytes B = MB × 1048576
  80. 11.80Megabytes → Kilobytes KB = MB × 1024
  81. 11.81Megabytes → Gigabytes GB = MB × 0.0009765625
  82. 11.82Gigabytes → Bytes B = GB × 1073741824
  83. 11.83Gigabytes → Kilobytes KB = GB × 1048576
  84. 11.84Gigabytes → Megabytes MB = GB × 1024
  85. 11.85Celsius → Fahrenheit °F = °C × 95 + 32
  86. 11.86Fahrenheit → Celsius °C = (°F − 32) × 59
  87. 11.87Celsius → Kelvin K = °C + 273.15
  88. 11.88Kelvin → Celsius °C = K − 273.15
  89. 11.89Fahrenheit → Kelvin K = (°F − 32) × 59 + 273.15
  90. 11.90Kelvin → Fahrenheit °F = (K − 273.15) × 95 + 32

12 Everyday & Health 8 topics

  1. 12.1Body Mass Index (BMI) BMI = mh2
  2. 12.2Basal Metabolic Rate 10m + 6.25h − 5a + s
  3. 12.3Daily Calorie Needs kcal = BMR · activity
  4. 12.4Speed (distance ÷ time) v = dt
  5. 12.5Fuel Economy (km/L) economy = distancefuel
  6. 12.6Unit Price unit price = totalquantity
  7. 12.7Running Pace (min/km) pace = timedistance
  8. 12.8Tip & Split per Person bill·(1+tip100)people
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