︠69be959a-2cb9-4eea-bd4a-4a02bcd5dec6︠ atan(1) * 4 ︡e68af1da-900a-4487-acc3-6af3790bd636︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\pi"}︡ ︠c945acae-1d95-49d6-aeed-deaa72b9daaf︠ float(atan(1) * 4.0) ︡d80332c6-a3f5-4eae-a556-57f7bd01abe0︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}3.14159265359"}︡ ︠a8878858-fed0-4a06-abc8-4dd5521fd153︠ solve((2*x)^2+x^2==81,x) ︡ced55a9c-1e25-43e4-b729-218f4345c556︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[x = -\\frac{9}{5} \\, \\sqrt{5}, x = \\frac{9}{5} \\, \\sqrt{5}\\right]"}︡ ︠178eb936-8f6a-4b09-bc46-33baca827636︠ show(solve((2*x)^2+x^2==81,x)) ︡b42e4d7e-79b7-4db9-9cd4-c90488464222︡{"html": "

\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[x = -\\frac{9}{5} \\, \\sqrt{5}, x = \\frac{9}{5} \\, \\sqrt{5}\\right]

"}︡ ︠fb2a9258-a01c-4ce8-a098-c056648efce7︠ a=var('a'); b=var('b'); c=var('c') show(solve(a*x^2+b*x+c==0,x)) ︡3c175bb9-c73b-499a-a7e0-8db18b0bc7cc︡{"stdout": "

\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[x = \\frac{-b + \\sqrt{-4 \\, a c + b^{2"}︡ ︠d1234340-b95d-4991-985f-dd746d534251i︠ %html 2 \, a}, x = \frac{-b - \sqrt{-4 \, a c + b^{2}}}{2 \, a}\right]

}}} ︡c89e7ace-9426-464f-bd7c-ddb2414f52c9︡{"html": "2 \\, a}, x = \\frac{-b - \\sqrt{-4 \\, a c + b^{2}}}{2 \\, a}\\right]\n}}}"}︡ ︠dfe79e4a-e188-48b1-9c42-e3aba1104250︠ maxima('solve(a*x^2+b*x+c=0,x)') ︡1a2b47f3-8fd5-4530-a9b8-cbe17ef59718︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[ x=-{{\\sqrt{b^2-4\\,a\\,c}+b}\\over{2\\,a}} , x={{\\sqrt{b^2-4\\,a \\,c}-b}\\over{2\\,a}} \\right] "}︡ ︠e792e2ef-e672-41cd-a146-66ffd9319b51︠ var('x,y,z') equ = [3*x + 7*y == 2, z*x + 3*y == 8] solve(equ,x,y) ︡27e7abf9-1728-4430-93d5-dccf6c541e7a︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[x = \\frac{50}{7 \\, z - 9}, y = \\frac{2 \\, {\\left(z - 12\\right)}}{7 \\, z - 9}\\right]\\right]"}︡ ︠ad5db99d-279e-4454-9957-aa3b1b129d97︠ f = maxima('1/sqrt(x^2+2*x-1)'); f ︡f470b7a4-67b9-4080-90e2-34416a7cda81︡{"stdout": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{{1}\\over{\\sqrt{x^2+2\\,x-1"}︡ ︠c24c7071-5c61-4e27-b33e-a52e16295cf9i︠ %html /span> }}} ︡c472ac4d-e1c3-4740-89ca-e1015409a346︡{"html": "/span>\n}}}"}︡ ︠610585f2-3cab-466f-8a7e-3938915de1df︠ type(f) ︡47cf7894-f3f1-4dc0-a099-b84e10aa1d36︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\hbox{ < class 'sage.interfaces.maxima.MaximaElement' > }"}︡ ︠5b32fdc5-d68b-4ba1-bed3-497d084e5fe2︠ f.integrate(x) ︡e1fe14ab-3b2f-4a15-92f4-b6f49e0e6727︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\log \\left(2\\,\\sqrt{x^2+2\\,x-1}+2\\,x+2\\right)"}︡ ︠ddd02d85-a213-474b-b54a-2af9bb995804︠ g = 1/sqrt(x^2+4*x-2); g ︡24386f8f-3a09-4453-b52a-10505c2d8d50︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{\\sqrt{x^{2} + 4 \\, x - 2}}"}︡ ︠6d401ad6-349a-4a59-8bc4-12f509cf7e91︠ type(g) ︡e72625a9-fb6c-434f-9a54-e7a9c587f443︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\hbox{ < type 'sage.symbolic.expression.Expression' > }"}︡ ︠badfaca7-a8b5-408a-a54f-9c768b813a27︠ g.integrate(x) ︡430c25fb-3f92-4f11-abac-d5ba20c2164f︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\log\\left(2 \\, x + 2 \\, \\sqrt{x^{2} + 4 \\, x - 2} + 4\\right)"}︡ ︠4ed0c159-ed91-461e-899b-dc23056bd59f︠ m = random_matrix(QQ,3,4); m ︡1e9ad2b5-7015-4ecd-abe4-c5c097b1d87a︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n-2 & 2 & 1 & 2 \\\\\n\\frac{1}{2} & 1 & 1 & -\\frac{1}{2} \\\\\n-\\frac{1}{2} & -1 & \\frac{1}{2} & 2\n\\end{array}\\right)"}︡ ︠12559dc7-34c2-4526-a6ca-84d5846fd2f9︠ m.echelon_form() ︡970b17fa-b145-4240-aacc-dd7f0aefc1d1︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rrrr}\n1 & 0 & 0 & -\\frac{4}{3} \\\\\n0 & 1 & 0 & -\\frac{5}{6} \\\\\n0 & 0 & 1 & 1\n\\end{array}\\right)"}︡ ︠926dafd6-d98f-41bf-a05e-5687c8141ff0︠ matrix(13, 1, [diff(sin(x)^2 + cos(x), i) for i in range(13)]) ︡af7f647b-0bf0-473b-851d-8a7deacc06f2︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{r}\n\\sin\\left(x\\right)^{2} + \\cos\\left(x\\right) \\\\\n2 \\, \\sin\\left(x\\right) \\cos\\left(x\\right) - \\sin\\left(x\\right) \\\\\n-2 \\, \\sin\\left(x\\right)^{2} + 2 \\, \\cos\\left(x\\right)^{2} - \\cos\\left(x\\right) \\\\\n-8 \\, \\sin\\left(x\\right) \\cos\\left(x\\right) + \\sin\\left(x\\right) \\\\\n8 \\, \\sin\\left(x\\right)^{2} - 8 \\, \\cos\\left(x\\right)^{2} + \\cos\\left(x\\right) \\\\\n32 \\, \\sin\\left(x\\right) \\cos\\left(x\\right) - \\sin\\left(x\\right) \\\\\n-32 \\, \\sin\\left(x\\right)^{2} + 32 \\, \\cos\\left(x\\right)^{2} - \\cos\\left(x\\right) \\\\\n-128 \\, \\sin\\left(x\\right) \\cos\\left(x\\right) + \\sin\\left(x\\right) \\\\\n128 \\, \\sin\\left(x\\right)^{2} - 128 \\, \\cos\\left(x\\right)^{2} + \\cos\\left(x\\right) \\\\\n512 \\, \\sin\\left(x\\right) \\cos\\left(x\\right) - \\sin\\left(x\\right) \\\\\n-512 \\, \\sin\\left(x\\right)^{2} + 512 \\, \\cos\\left(x\\right)^{2} - \\cos\\left(x\\right) \\\\\n-2048 \\, \\sin\\left(x\\right) \\cos\\left(x\\right) + \\sin\\left(x\\right) \\\\\n2048 \\, \\sin\\left(x\\right)^{2} - 2048 \\, \\cos\\left(x\\right)^{2} + \\cos\\left(x\\right)\n\\end{array}\\right)"}︡ ︠ae24bf14-cbc1-4915-a598-f526b65de55e︠ factor(2012) ︡0de2590d-4ffe-4341-86ba-fadb9eaf8840︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}2^{2} \\cdot 503"}︡ ︠6e2bab38-74bb-463e-ab8e-9f474c7af396︠ d = next_prime(1000000000000); d ︡38f1c94a-5935-4c28-81ec-04ffe9ce083e︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}1000000000039"}︡ ︠31e4386b-823e-4d7d-91f6-344c38c4d557︠ factor(d) ︡a4751777-98f2-45ee-842a-6de7b611d8e6︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}1000000000039"}︡ ︠97befb76-73a4-4d31-b15b-3ff1504ed926︠ e = next_prime(45217845633476111); e ︡843949a9-841a-406a-b4fb-bc1d70f76280︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}45217845633476177"}︡ ︠c80c1588-71eb-4868-8eac-b9371a2446c6︠ f = (d*e)+1; f ︡8d45df20-9ed2-467c-b2d3-26dfd9f7ef93︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}45217845635239672979705570904"}︡ ︠adb8707e-17d1-41d9-ba08-6d357a61c338︠ factor(f) ︡8ef41663-e71c-4a70-b993-d4144f56eab3︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}2^{3} \\cdot 3 \\cdot 7 \\cdot 13 \\cdot 2064913 \\cdot 10026641206858177387"}︡ ︠5c89896e-546b-4029-a705-aa5b43ae2daf︠ is_prime(10026641206858177387) ︡9c913059-53be-4f37-9223-453881c2c98c︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\rm True}"}︡ ︠2bbccc6f-1d1f-4951-9463-b7a8c466d314︠ factor((2*x)^2+x) ︡f71cf3ac-31c6-4a04-8610-b358a9a96d56︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(4 \\, x + 1\\right)} x"}︡ ︠4e4348f7-e4c4-49e0-bcd3-ed2e38040e85︠ x, y = var('x, y'); factor(x^3 - sin(y)^3) ︡5b3bd672-2b06-4072-aaac-265a27c51c70︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(x - \\sin\\left(y\\right)\\right)} {\\left(x^{2} + x \\sin\\left(y\\right) + \\sin\\left(y\\right)^{2}\\right)}"}︡ ︠8f814458-b8e1-4ed1-a75a-e008efb57294︠ F. = GF(49) x = polygen(F) factor(x^4 + x^3 - 2) ︡d0c064b6-7b7e-44dc-b3f0-93e30ede83c8︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}(x + \\alpha + 1) \\cdot (x + 6 \\alpha + 2) \\cdot (x + 6)^{2}"}︡ ︠9f6e340e-00d0-4512-ac1b-24bcabce3280︠ type(F) ︡daea1fc7-ed8a-4d91-82e5-dd3ec903f083︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\hbox{ < class 'sage.rings.finite_rings.finite_field_givaro.FiniteField_givaro' > }"}︡ ︠62570df3-a9c7-4e63-b04d-9136ad24277a︠ var('x,y,z') solve([3*x + 7*y == 2, z*x + 3*y == 8],x,y) ︡a5b87b46-16a7-47c8-bc8d-f402a625e75d︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[\\left[x = \\frac{50}{7 \\, z - 9}, y = \\frac{2 \\, {\\left(z - 12\\right)}}{7 \\, z - 9}\\right]\\right]"}︡ ︠b877b852-215c-412e-a4ec-e618c63e3261︠ var('alpha') A = matrix([ [3, 7], [alpha, 3] ]) A ︡e35e12eb-c6e1-4a06-96df-ecfdd80bf976︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\begin{array}{rr}\n3 & 7 \\\\\n\\alpha & 3\n\\end{array}\\right)"}︡ ︠bca85b55-5b88-40fd-a7d9-850af9b60f14︠ v = vector([2,8]) v ︡011445a9-6d47-44b3-92c3-358c09449549︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(2,8\\right)"}︡ ︠4ce5d1c9-9fb7-4dc8-80af-2eb1bce95115︠ u = A.solve_right(v) u ︡6a6b4853-43f2-4f0c-916e-863207768d23︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left(\\frac{-14 \\, {\\left(\\alpha - 12\\right)}}{3 \\, {\\left(7 \\, \\alpha - 9\\right)}} + \\frac{2}{3},\\frac{2 \\, {\\left(\\alpha - 12\\right)}}{7 \\, \\alpha - 9}\\right)"}︡ ︠b00d182b-ccfc-459d-a375-832d272a6b48︠ integrate(sin(x)*tan(x),x) ︡68256e5f-a445-420f-b599-52da26f0aa3e︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}-\\frac{1}{2} \\, \\log\\left(\\sin\\left(x\\right) - 1\\right) + \\frac{1}{2} \\, \\log\\left(\\sin\\left(x\\right) + 1\\right) - \\sin\\left(x\\right)"}︡ ︠3c65b978-aa09-476a-b8c9-906c97957084︠ diff(sin(x)*tan(x)) ︡64436488-2199-4760-bbb7-f0f2a3220dc9︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left(\\tan\\left(x\\right)^{2} + 1\\right)} \\sin\\left(x\\right) + \\cos\\left(x\\right) \\tan\\left(x\\right)"}︡ ︠e0afe291-7a58-47e5-97a4-2d4cf23b77c4︠ sin(x)*tan(x).simplify_full() ︡33a69cd6-1ac0-486e-92f7-52b1daa76181︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{\\sin\\left(x\\right)^{2}}{\\cos\\left(x\\right)}"}︡ ︠52e2cae1-a550-429f-a337-53b197d23e0e︠ fn = 1/sqrt(x^2 + 2*x - 1); fn ︡c218c547-a18b-4ae7-bf7d-6399e3fdbd56︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\frac{1}{\\sqrt{x^{2} + 2 \\, x - 1}}"}︡ ︠8386391c-01a5-4bc4-84fd-a784d2a415e8︠ plot(fn, (x, .5 ,2), gridlines=True) ︡4b01fc6f-80e2-4ee8-b2c5-6c79df124a8c︡{"html": "

"}︡ ︠eaeba83d-4c75-450d-99af-e6a1717ef33f︠ var('x,y') plot3d(sin(x-y)*y*cos(x), (x,-3,3), (y,-3,3), opacity=.75) ︡0f51beda-e083-4aef-965b-23af4a302736︡︡ ︠b3313a9b-dcfc-4eee-a541-b963d779724c︠ gn = (2*x)^2+x^2==81; gn ︡a806722b-5c6b-467d-845a-57f14ec8e9cc︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}5 \\, x^{2} = 81"}︡ ︠e3a6b419-9f95-415b-a1bf-ef211e6fcc20︠ plot(gn, (x, -5 ,5), gridlines=True, fill=True) ︡5dc15987-af6c-4a45-959b-7e04134932b9︡{"html": "

"}︡ ︠483548a9-df53-4356-ab00-89f046e4a53c︠ plot3d(gn, (x,-5,5), (y,-5,5), opacity=.75) ︡6799cd84-1a09-40f1-9d9d-9bf0d28bba71︡︡ ︠c1664902-507d-4c66-9473-7318dfe42f6d︠ plot(sin(x), (x, -pi ,pi), gridlines=True, fill=True) ︡0ffe12fc-3af0-43b9-8e1c-3164b6e9c97a︡{"html": "

"}︡ ︠3a47e1d5-cba4-43b1-9665-e0c61e12ec9a︠ u, v = var('u,v') f_x = cos(u)*(4*sqrt(1-v^2)*sin(abs(u))^abs(u)) f_y = sin(u) *(4*sqrt(1-v^2)*sin(abs(u))^abs(u)) f_z = v parametric_plot3d([f_x, f_y, f_z], (u, -pi, pi), (v, -1, 1), frame=False, color="red") ︡f9d478a9-c755-48f6-bba9-732b278d4e6f︡︡ ︠d9d9808e-48e0-4c79-a1b4-47adf609caf0︠ eqn = (2*x)^2 == -x^2 + 81 eqn ︡7d315f53-df11-4449-b671-046b19b6714e︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}4 \\, x^{2} = -x^{2} + 81"}︡ ︠6fb70981-6c86-477c-83f0-34563d381fdf︠ eqn = eqn + x^2; eqn ︡fb4fe4a4-dd30-473e-b4ba-8462ea34ae0c︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}5 \\, x^{2} = 81"}︡ ︠543e0a9e-4acc-4ec3-b9c3-9a7bdd3413eb︠ eqn = eqn / 5; eqn ︡f59181fc-f257-47eb-8add-ec47f67d185c︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}x^{2} = \\left(\\frac{81}{5}\\right)"}︡ ︠20b2f011-10b6-4fbb-a0bc-7b6a5d852813︠ eqn = sqrt(eqn); eqn ︡87cd1d91-6a7b-493d-961c-b11c2a333996︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\sqrt{x^{2} = \\left(\\frac{81}{5}\\right)}"}︡ ︠91082eaa-d756-44c3-afe9-de0fd1ce2b34︠ eqn.simplify_full() ︡2569d4e6-7e54-4550-96ac-e0968b1374f7︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}{\\left| x \\right|} = \\frac{9}{5} \\, \\sqrt{5}"}︡ ︠af3f4368-3c70-4cec-9ca9-ca7573e52512︠ plot(sin, (0,2*pi), plot_points=2000) ︡aecd5bfe-e737-4cee-8ade-6734fd04b29a︡{"html": "

"}︡ ︠b1d63d6a-3d00-4bae-aba2-ff8406ef6146︠ def g(f): return plot(sin(2*x*pi*f),(x,0,2*pi)) @interact def _(f=(.1,1,.1)): show(g(f)) ︡f5a981a5-c748-4ab5-a15f-49aa99c881ac︡︡ ︠dfa00b5a-383c-4737-a850-00754a1755a2︠ [random() for i in [1 .. 5]] ︡2dffa77e-ebe8-4159-96f7-8106af7398a2︡{"html": "\\newcommand{\\Bold}[1]{\\mathbf{#1}}\\left[0.721797750678, 0.282854097694, 0.192115487542, 0.399790287024, 0.480238455049\\right]"}︡