You are watching: What is the reciprocal of 0
I do recognize that $frac10$ is technically undefined and infinity can"t really be treated like an additional number, but could they be reciprocals in some situations?
So, is $frac1infty$ some infinitesimal, or is that zero? and does $frac10=infty$?
I"m i m really sorry if this is a stupid and also obvious question, my knowledge of historicsweetsballroom.com in the realm of infinity is... Shaky to say the least.
station infinity riemann-sphere
edited Feb 25 "19 in ~ 3:49
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asked Feb 25 "19 in ~ 2:53
Rory M. TimsRory M. Tims
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It depends on the number system you"re using.
If you"re making use of the genuine numbers or the complex numbers, then zero has no reciprocal. In various other words, $1/0$ is an undefined expression. Also, in this systems, there"s no such number together infinity. In other words, $infty$ is an undefined symbol.
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If you"re utilizing the projectively expanded real heat or the Riemann sphere, climate the mutual of zero is infinity, and also the mutual of infinity is zero. In other words, $1/0 = infty$ and also $1 / infty = 0$. (Note that the mutual of infinity is specifically zero, no infinitesimal. No one of these 4 number solution contain any kind of infinitesimal numbers.)
Out the these four number systems, the first two (the real numbers and the complex numbers) space much an ext commonly used than the last two (the projectively prolonged real line and also the Riemann sphere). So much so, in fact, the we commonly say "division by $0$ is undefined" and "infinity is not a number" there is no clarifying which device we"re using.
The factor that the an initial two solution are more commonly offered is that these 2 systems are fields, and also the various other two room not.