模組:沙盒/a2569875/NumberTest/doc

• ${{2}\times {4}}=8$

$\times$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $1$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $2$ $2$ $4$ $6$ $8$ $10$ $12$ $14$ $16$ $18$ $3$ $3$ $6$ $9$ $12$ $15$ $18$ $21$ $24$ $27$ $4$ $4$ $8$ $12$ $16$ $20$ $24$ $28$ $32$ $36$ $5$ $5$ $10$ $15$ $20$ $25$ $30$ $35$ $40$ $45$ $6$ $6$ $12$ $18$ $24$ $30$ $36$ $42$ $48$ $54$ $7$ $7$ $14$ $21$ $28$ $35$ $42$ $49$ $56$ $63$ $8$ $8$ $16$ $24$ $32$ $40$ $48$ $56$ $64$ $72$ $9$ $9$ $18$ $27$ $36$ $45$ $54$ $63$ $72$ $81$	$+$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $4$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $5$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $6$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $7$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $16$ $8$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $9$ $10$ $11$ $12$ $13$ $14$ $15$ $16$ $17$ $18$	$-$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $1$ $0$ $-1$ $-2$ $-3$ $-4$ $-5$ $-6$ $-7$ $-8$ $2$ $1$ $0$ $-1$ $-2$ $-3$ $-4$ $-5$ $-6$ $-7$ $3$ $2$ $1$ $0$ $-1$ $-2$ $-3$ $-4$ $-5$ $-6$ $4$ $3$ $2$ $1$ $0$ $-1$ $-2$ $-3$ $-4$ $-5$ $5$ $4$ $3$ $2$ $1$ $0$ $-1$ $-2$ $-3$ $-4$ $6$ $5$ $4$ $3$ $2$ $1$ $0$ $-1$ $-2$ $-3$ $7$ $6$ $5$ $4$ $3$ $2$ $1$ $0$ $-1$ $-2$ $8$ $7$ $6$ $5$ $4$ $3$ $2$ $1$ $0$ $-1$ $9$ $8$ $7$ $6$ $5$ $4$ $3$ $2$ $1$ $0$
$/$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $1$ $1$ $0.5$ $0.33333333333333$ $0.25$ $0.2$ $0.16666666666667$ $0.14285714285714$ $0.125$ $0.11111111111111$ $2$ $2$ $1$ $0.66666666666667$ $0.5$ $0.4$ $0.33333333333333$ $0.28571428571429$ $0.25$ $0.22222222222222$ $3$ $3$ $1.5$ $1$ $0.75$ $0.6$ $0.5$ $0.42857142857143$ $0.375$ $0.33333333333333$ $4$ $4$ $2$ $1.3333333333333$ $1$ $0.8$ $0.66666666666667$ $0.57142857142857$ $0.5$ $0.44444444444444$ $5$ $5$ $2.5$ $1.6666666666667$ $1.25$ $1$ $0.83333333333333$ $0.71428571428571$ $0.625$ $0.55555555555556$ $6$ $6$ $3$ $2$ $1.5$ $1.2$ $1$ $0.85714285714286$ $0.75$ $0.66666666666667$ $7$ $7$ $3.5$ $2.3333333333333$ $1.75$ $1.4$ $1.1666666666667$ $1$ $0.875$ $0.77777777777778$ $8$ $8$ $4$ $2.6666666666667$ $2$ $1.6$ $1.3333333333333$ $1.1428571428571$ $1$ $0.88888888888889$ $9$ $9$ $4.5$ $3$ $2.25$ $1.8$ $1.5$ $1.2857142857143$ $1.125$ $1$	$mod$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $1$ $0$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $0$ $0$ $2$ $2$ $2$ $2$ $2$ $2$ $2$ $3$ $0$ $1$ $0$ $3$ $3$ $3$ $3$ $3$ $3$ $4$ $0$ $0$ $1$ $0$ $4$ $4$ $4$ $4$ $4$ $5$ $0$ $1$ $2$ $1$ $0$ $5$ $5$ $5$ $5$ $6$ $0$ $0$ $0$ $2$ $1$ $0$ $6$ $6$ $6$ $7$ $0$ $1$ $1$ $3$ $2$ $1$ $0$ $7$ $7$ $8$ $0$ $0$ $2$ $0$ $3$ $2$ $1$ $0$ $8$ $9$ $0$ $1$ $0$ $1$ $4$ $3$ $2$ $1$ $0$	$x^{y}$ $1$ $2$ $3$ $4$ $5$ $6$ $7$ $8$ $9$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $1$ $2$ $2$ $4$ $8$ $16$ $32$ $64$ $128$ $256$ $512$ $3$ $3$ $9$ $27$ $81$ $243$ $729$ $2187$ $6561$ $19683$ $4$ $4$ $16$ $64$ $256$ $1024$ $4096$ $16384$ $65536$ $262144$ $5$ $5$ $25$ $125$ $625$ $3125$ $15625$ $78125$ $390625$ $1953125$ $6$ $6$ $36$ $216$ $1296$ $7776$ $46656$ $279936$ $1679616$ $10077696$ $7$ $7$ $49$ $343$ $2401$ $16807$ $117649$ $823543$ $5764801$ $40353607$ $8$ $8$ $64$ $512$ $4096$ $32768$ $262144$ $2097152$ $16777216$ $134217728$ $9$ $9$ $81$ $729$ $6561$ $59049$ $531441$ $4782969$ $43046721$ $387420489$

Dual number

• 1+0.5ε

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• ${{\varepsilon }^{0}}=1$
• ${{\varepsilon }^{1}}=\varepsilon$
• ${{\varepsilon }^{2}}=0$
• ${{\varepsilon }^{3}}=0$

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• ${\sqrt {{1}+{\varepsilon }}}=1+0.5\varepsilon$
• ${\sqrt {{0.1}+{\varepsilon }}}=0.31622776601684+1.5811388300842\varepsilon$
• ${\sqrt {{0.01}+{\varepsilon }}}=0.1+5\varepsilon$
• ${\sqrt {{0.001}+{\varepsilon }}}=0.031622776601684+15.811388300842\varepsilon$
• ${\sqrt {{{1}\times {{10}^{-{7}}}}+{\varepsilon }}}=0.00031622776601684+1581.1388300842\varepsilon$
• ${\sqrt {{{1}\times {{10}^{-{10}}}}+{\varepsilon }}}=1\times 10^{-05}+50000\varepsilon$

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• $\left({{1}+{\varepsilon }}\right)^{-1}=1-\varepsilon$
• $\left({{0.1}+{\varepsilon }}\right)^{-1}=10-100\varepsilon$
• $\left({{0.01}+{\varepsilon }}\right)^{-1}=100-10000\varepsilon$
• $\left({{0.001}+{\varepsilon }}\right)^{-1}=1000-1000000\varepsilon$
• $\left({{{1}\times {{10}^{-{7}}}}+{\varepsilon }}\right)^{-1}=10000000-1\times 10^{+14}\varepsilon$
• $\left({{{1}\times {{10}^{-{10}}}}+{\varepsilon }}\right)^{-1}=10000000000-1\times 10^{+20}\varepsilon$

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• ${\sqrt {{1}+{\varepsilon }}}=1+0.5\varepsilon$
• ${{\sqrt {{1}+{\varepsilon }}}\,{\sqrt {{1}+{\varepsilon }}}}=1+\varepsilon$
• $e^{{2}+{{3}\,{\varepsilon }}}=7.3890560989307+22.167168296792\varepsilon$
• $\ln e^{{2}+{{3}\,{\varepsilon }}}=2+3\varepsilon$
• ${{\left({{2}+{{3}\,{\varepsilon }}}\right)}^{{5}+{{7}\,{\varepsilon }}}}=32+395.26496844543\varepsilon$
• $\log _{{2}+{{3}\,{\varepsilon }}}\left({{\left({{2}+{{3}\,{\varepsilon }}}\right)}^{{5}+{{7}\,{\varepsilon }}}}\right)=5+7\varepsilon$
• $\left({{5}+{{7}\,{\varepsilon }}}\right)^{-1}=0.2-0.28\varepsilon$
• ${{\left({{5}+{{7}\,{\varepsilon }}}\right)}\,{\left({{5}+{{7}\,{\varepsilon }}}\right)^{-1}}}=1$
• $\sin \left({{1.5}+{{2}\,{\varepsilon }}}\right)=0.99749498660405+0.14147440333541\varepsilon$
• $\arcsin \sin \left({{1.5}+{{2}\,{\varepsilon }}}\right)=1.5+2\varepsilon$
• $\arccos \left({{0.5}+{{\pi }\,{\varepsilon }}}\right)=1.0471975511966-3.6275987284684\varepsilon$
• $\cos \arccos \left({{0.5}+{{\pi }\,{\varepsilon }}}\right)=0.5+3.1415926535898\varepsilon$

$\times$	$1$	$\varepsilon$
$1$	$1$	$\varepsilon$
$\varepsilon$	$\varepsilon$	$0$

$\times$	$1$	$i$	$\varepsilon$	$i\varepsilon$
$1$	$1$	$i$	$\varepsilon$	$i\varepsilon$
$i$	$i$	$-1$	$i\varepsilon$	$-\varepsilon$
$\varepsilon$	$\varepsilon$	$-i\varepsilon$	$0$	$0$
$i\varepsilon$	$i\varepsilon$	$\varepsilon$	$0$	$0$

$\times$	$1$	$i$	$\varepsilon j$	$\varepsilon k$
$1$	$1$	$i$	$\varepsilon j$	$\varepsilon k$
$i$	$i$	$-1$	$\varepsilon k$	$-\varepsilon j$
$\varepsilon j$	$\varepsilon j$	$-\varepsilon k$	$0$	$0$
$\varepsilon k$	$\varepsilon k$	$\varepsilon j$	$0$	$0$