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Category: Number Theory

Show-that-e-ipi-1-0-

Question Number 8838 by tawakalitu last updated on 31/Oct/16 $$\mathrm{Show}\:\mathrm{that}\::\:\:\mathrm{e}^{\mathrm{i}\pi\:+\:\mathrm{1}} \:=\:\mathrm{0} \\ $$ Answered by FilupSmith last updated on 31/Oct/16 $$\mathrm{do}\:\mathrm{you}\:\mathrm{mean}\:{e}^{{i}\pi} +\mathrm{1}=\mathrm{0} \\ $$$${e}^{{ix}} =\mathrm{cos}\left({x}\right)+{i}\mathrm{sin}\left({x}\right)…

Question-139668

Question Number 139668 by aupo14 last updated on 30/Apr/21 Commented by mr W last updated on 30/Apr/21 $${x}^{\mathrm{2}} \:{can}\:{only}\:{be}\:{formed}\:{from}\:{two}\:{times}\: \\ $$$$\left(\mathrm{1}+{x}+{x}^{\mathrm{3}} \right).\:{therefore}\:{the}\:{coefficient} \\ $$$${of}\:{x}^{\mathrm{2}} \:{is}\:{C}_{\mathrm{2}}…

n-1-2n-2-3n-3-rn-r-n-n-1-n-1-n-2-n-3-n-r-1-n-1-2-n-2-3-n-3-4-n-4-r-n-r-1-0-n-1-n-2-n-3-n-Prove-the-above-identity-

Question Number 139560 by Dwaipayan Shikari last updated on 28/Apr/21 $$\underset{{n}_{\mathrm{1}} +\mathrm{2}{n}_{\mathrm{2}} +\mathrm{3}{n}_{\mathrm{3}} +..+{rn}_{{r}} ={n}} {\overset{{n}} {\sum}}\frac{\mathrm{1}}{{n}_{\mathrm{1}} !{n}_{\mathrm{2}} !{n}_{\mathrm{3}} !..{n}_{{r}} !\mathrm{1}^{{n}_{\mathrm{1}} } \mathrm{2}^{{n}_{\mathrm{2}} } \mathrm{3}^{{n}_{\mathrm{3}}…

Determine-smallest-n-0-for-which-i-n-1-

Question Number 8336 by Rasheed Soomro last updated on 08/Oct/16 $$\mathrm{Determine}\:\mathrm{smallest}\:\mathrm{n}\left(\neq\mathrm{0}\right),\:\mathrm{for}\:\mathrm{which} \\ $$$$\left(\omega+\mathrm{i}\right)^{\mathrm{n}} =\mathrm{1}. \\ $$ Commented by prakash jain last updated on 08/Oct/16 $${w}=\mathrm{cos}\:\frac{\mathrm{2}\pi}{\mathrm{3}}+{i}\mathrm{sin}\:\frac{\mathrm{2}\pi}{\mathrm{3}}=−\frac{\mathrm{1}}{\mathrm{2}}+{i}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}…

is-it-correct-when-S-n-1-2-1-3-1-4-1-5-1-n-2-3-2-4-2-5-2-n-3-4-3-5-3-n-4-5-4-n-

Question Number 8139 by sou1618 last updated on 01/Oct/16 $${is}\:{it}\:{correct}? \\ $$$${when} \\ $$$${S}_{{n}} =\left\{\right. \\ $$$$\mathrm{1}×\mathrm{2}+\mathrm{1}×\mathrm{3}+\mathrm{1}×\mathrm{4}+\mathrm{1}×\mathrm{5}+…….+\mathrm{1}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:+\mathrm{2}×\mathrm{3}+\mathrm{2}×\mathrm{4}+\mathrm{2}×\mathrm{5}+…….+\mathrm{2}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{3}×\mathrm{4}+\mathrm{3}×\mathrm{5}+…….+\mathrm{3}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{4}×\mathrm{5}+…….+\mathrm{4}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….…

Given-that-Z-and-H-are-complex-number-obtain-the-real-and-imaginary-of-Z-H-

Question Number 7748 by Tawakalitu. last updated on 13/Sep/16 $${Given}\:{that}\:{Z}\:{and}\:{H}\:{are}\:{complex}\:{number}.\: \\ $$$${obtain}\:{the}\:{real}\:{and}\:{imaginary}\:{of}\:{Z}^{{H}} \\ $$ Answered by Yozzia last updated on 13/Sep/16 $${Let}\:{Z}={re}^{{i}\theta} ,\:{H}={c}+{di}\:\:\left({r},\theta,{c},{d}\in\mathbb{R},\:{r}>\mathrm{0},\:{i}=\sqrt{−\mathrm{1}}\right). \\ $$$${Z}^{{H}}…