2200/9

2200/9

Input 2200/9 Exact result 2200/9 (irreducible) Decimal approximation 244.4444444444444444444444444444444444444444444444444444444… Repeating decimal 244.4^_ (period 1) Mixed fraction 244 4/9 Quotient and remainder 244×9 + 4 Prime factorization 2^3×3^(-2)×5^2×11 Continued fraction [244; 2, 4] Egyptian fraction expansion 244 + 1/3 + 1/9

2cosx-1

2cosx-1

Input 2 cos(x) – 1 Alternate form e^(-i x) + e^(i x) – 1 Series expansion at x=0 1 – x^2 + x^4/12 + O(x^6) (Taylor series) Derivative d/dx(2 cos(x) – 1) = -2 sin(x) Indefinite integral integral (-1 + 2 cos(x)) dx = 2 sin(x) – x + constant Global maxima max{2 cos(x) – 1} = 1 at x = 2 pi n for integer n Definite integral over a half-period integral_0^pi (-1 + 2 cos(x)) dx = -pi~~-3.14159…

Read More Read More

3 8 7 8

3 8 7 8

Input {3, 8, 7, 8} Total 3 + 8 + 7 + 8 = 26 Statistics mean | 6.5 median | 7.5 sample standard deviation | 2.38 Vector length sqrt(186)~~13.6382 Normalized vector (sqrt(3/62), 4 sqrt(2/93), 7/sqrt(186), 4 sqrt(2/93)) Possible closed form (no form found in terms of holonomic sequences)

2ln5

2ln5

Input 2 log(5) Decimal approximation 3.218875824868200749201518666452375279051202708537035443825… Property 2 log(5) is a transcendental number Alternate form log(25) Continued fraction [3; 4, 1, 1, 3, 7, 2, 12, 47, 2, 1, 1, 1, 1, 1, 1, 19, 1, 11, 1, 5, 15, 3, 1, 1, 3, 1, 1, 19, …]

2cotx

2cotx

Input 2 cot(x) Alternate form assuming x is real -(2 sin(2 x))/(cos(2 x) – 1) Roots x = 1/2 (2 pi n + pi), n element Z Series expansion at x=0 2/x – (2 x)/3 – (2 x^3)/45 – (4 x^5)/945 + O(x^6) (Laurent series) Derivative d/dx(2 cot(x)) = -2 csc^2(x) Indefinite integral integral 2 cot(x) dx = 2 log(sin(x)) + constant

216/125

216/125

Input 216/125 Exact result 216/125 (irreducible) Decimal form 1.728 Mixed fraction 1 91/125 Quotient and remainder 1×125 + 91 Prime factorization 2^3×3^3×5^(-3) Continued fraction [1; 1, 2, 1, 2, 11] Egyptian fraction expansion 1 + 1/2 + 1/5 + 1/36 + 1/4500