54 lines
1.3 KiB
Plaintext
54 lines
1.3 KiB
Plaintext
Defining a "real number"
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Natural Numbers (granted)
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1 2 3 4 5 ...
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Integers?
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"I want to close my number set under the operation of subtraction."
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(a1, a2) - (b1, b2) = (a1 + b2, a2 + b1)
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a = (3, 4); b = (5, 6);
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(a1, a2) equals (b1, b2) if a1 + b2 = a2 + b1
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Rationals?
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"I want to close my number set under the operation of division."
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(a1, a2) / (b1, b2) = (a1*b2, a2*b1)
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(a1, a2) equals (b1, b2) if a1*b2 = a2*b1
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Reals?
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My examples of definitions that aren't solid:
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~ Anything/any point you can plot on the number line
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~ Dedekind (spelling?) cut
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"I want to close my number set under limits."
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By limit I mean the n-ε-definition
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(a1, a2, a3, a4, ...)
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( 1, 0, 1, 0, ...)
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(1/2, 1/2, 1/2, 1/2, ...)
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(1/2, 3/4, 7/8, 15/16, ...)
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Start with the range (1,2)
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Bisect the range - x - this is the next element of the series
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Square x
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if (x > 2) update the range to (lower,x)
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else update the range to (x,upper)
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limit (3/2, 5/4, 11/8, 21/16, ...) = y
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Cauchy sequence - a sequence given any ε there is an n
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such that for any m,r >= n |am - ar| < ε
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"Close under limits." means
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"We want a set of numbers where every Cauchy sequence converges
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to a number in the set."
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(a1, a2, a3, a4, ...)
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a + b = (a1, a2, a3, a4, ...) + (b1, b2, b3, b4, ...)
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= (a1 + b1, a2 + b2, a3 + b3, ...) |