The word

Counting numbers are will now also be called

E is an event set.

E = (A, B, [C ::

A = Q is a set of rational numbers, and z is a rational number equal to zero.

B = If x is in Q, then z becomes the sum of x and the previous value of z, and x is removed from Q.

C = If Q is not empty, repeat B; otherwise continue to D.

D = Stop.

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Saying that "z is a rational number" implies, from the definition of

C is a counting set containing x, y and z.

Q is a counting set containing x and y.

Each element in Q is in C.

The unit string size of C is d, and the unit string size of Q is e, and d = e

If a is associated with b in Q, then e elements in C are between a and b in C.

Zero in C is associated with w in C.

Element p in C is associated with 0 in C.

S is a set.

Each element in S is a set.

The intersection of all of the sets in S is empty.

If v is a set in S, then the string from w to the absolute value of x in C has a one-to-one correspondence with v.

S has a one-to-one correspondence with the string from w to the absolute value of y in Q.

If x and y are of the same sign, the string from w to z in C has a one-to-one correspondence with the union of all the sets in S.

If x and y are not of the same sign, the string from z to p in C has a one-to-one correspondence with the union of all the sets in S.

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Q is a subset of C. The numbers to be multiplied must be chosen from the subset Q of C in order to insure that the product is in C. If the numbers to be multiplied are chosen from a counting set Q having a unit string size of ten, the product may not be in Q. For example, 0.2 ⋅ 0.3 = 0.06. The product is an element in a counting set having a unit string size of 100. The product of two rational numbers chosen from any counting set is always an element in a counting set of some unit string size. In fact, it is always an element of a counting set that has a unit string size that is the square of the original. The product may not be in the original subset, but it is no less a rational number because it is an element in a correctly defined counting set.

While they form a rigorous foundation, these technicalities will not add to the difficulty of calculating. The length of the string of numerals to the right of the decimal point is routinely modified to fit the precision of the calculation. Now, it is hoped, that the practice has a development in terms of sets.

E is an event set.

E = (A, B, [C ::

A = Rational number z = 0, and Q is a set of rational numbers;

B = If x is in Q, then z becomes the product of x and the previous value of z, and x is removed from Q.

C = If Q is not empty, repeat B; otherwise continue to D.

D = Stop.

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Each element in S is an integer equal to P

The product of all of the integers in S equals N.

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Notice that any quotient p/q would be wholly ambiguous if q were allowed to be zero. Such would-be ratios are considered undefined.

E is an event set.

E = (A, B, [C ::

A = Q is a counting set containing positive integer x and rational number y; and z in Q equals 0, and K is a non-empty set containing x elements.

B = If any element q is in K, then z becomes the product of y and the previous value of z, and q is removed from K.

C = If K is not empty, repeat B; otherwise continue to D.

D = Stop.

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R = the absolute value of x.

T = y

Z = 1/T.

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All ratios p/q where p and q are integers and q ≠ 0 can be named by means of some counting set. Notice that these definitions define y

4

4

2

The square root of 8 in the last example can only be approximated by rational numbers. It can, however, be approximated to any desired precision in any base. This is also true of π (the circumference of a circle divided by its diameter). This issue will be addressed later in this development.