48 lines
1.9 KiB
Plaintext
48 lines
1.9 KiB
Plaintext
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Addition: Multiplication:
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Infinity + Finite = Infinity Infinity x Infinity = Infinity
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Infinity + Infinity = Infinity Infinity x Finite = Infinity,
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but Infinity x 0 is undefined
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Infinity + -Infinity can be
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absolutely anything finite or not Infinity x -Infinity = -Infinity
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-Infinity + Finite = -Infinity -Infinity x Finite = -Infinity,
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with the same exception for 0 as before
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-Infinity + -Infinity = -Infinity
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-Infinity x -Infinity = Infinity
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Subtraction:
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Same as addition, with u-v treated as u+(-v):
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where
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-(Infinity) = -Infinity
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-(-Infinity) = Infinity
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Division:
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Same as multiplication, with u/v treated as u x (1/v):
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where
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1/(-Infinity) = -0
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1/(Infinity) = +0
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1/(-0) = -Infinity
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1/(+0) = Infinity
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You'll need to make the distinction between +0 and -0, if you're going to say
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anything useful about division with infinity.
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These rules are made in such a way that all the properties (+,x,-,/) will
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remain true when infinite limits are included. It is possible for a limit
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to be infinite without its positive or negative sign being determined. This
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limit will represent the unsigned infinity. Its negative is itself and its
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reciporical is 0 (without the + or - sign). You'll need to use all three
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kinds of infinity. Much of Calculus is devoted to resolving those limits
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involving the undefined operations above, like
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Infinity - Infinity, Infinity x 0, Infinity/Infinity
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There is a theory of infinitesimals based on what is known as Non-Standard
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Analysis. Its content is completely equivalent to Calculus. In fact, it is
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a reformulation of Calculus that matches very closely the original formulation
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of Calculus as a calculation system for infinite and infinitesimal numbers.
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