Defining a "real number"

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Natural Numbers (granted)
1 2 3 4 5 ...

Integers?
"I want to close my number set under the operation of subtraction."

(a1, a2) - (b1, b2) = (a1 + b2, a2 + b1)

a = (3, 4); b = (5, 6);

(a1, a2) equals (b1, b2) if a1 + b2 = a2 + b1

Rationals?
"I want to close my number set under the operation of division."

(a1, a2) / (b1, b2) = (a1*b2, a2*b1)

(a1, a2) equals (b1, b2) if a1*b2 = a2*b1

Reals?
My examples of definitions that aren't solid:
~ Anything/any point you can plot on the number line
~ Dedekind (spelling?) cut

"I want to close my number set under limits."
By limit I mean the n-ε-definition

(a1, a2, a3, a4, ...)

( 1,  0,  1,  0, ...)
(1/2, 1/2, 1/2, 1/2, ...)
(1/2, 3/4, 7/8, 15/16, ...)

Start with the range (1,2)
Bisect the range - x - this is the next element of the series
Square x
if (x > 2) update the range to (lower,x)
else update the range to (x,upper)
limit (3/2, 5/4, 11/8, 21/16, ...) = y

Cauchy sequence - a sequence given any ε there is an n
such that for any m,r >= n |am - ar| < ε

"Close under limits." means
"We want a set of numbers where every Cauchy sequence converges
to a number in the set."

(a1, a2, a3, a4, ...)

a + b = (a1, a2, a3, a4, ...) + (b1, b2, b3, b4, ...)
      = (a1 + b1, a2 + b2, a3 + b3, ...)