The numeral 1 is a very special text item. It represents the term one as an amount of things (a single thing).
Further, if initially a set, V, contains n elements, and then a single element is inserted in set V, the amount of things in V becomes a single thing more than n things, or one thing more than n things, and the new amount of things in V is named by the whole number next in order from n in the set of whole numbers, W.
This means that the whole number next in order from n is 1 more than n.
Definition - Item e is appended to bounded ordered set S if and only if each following statement is true:
S is a bounded ordered set.
The last element in S is d. Item e is inserted into S such that e is next order from d, and thereby becomes the new last element in S.
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Definition - Last element z is removed from bounded ordered set S if and only if each following statement is true:
S is a non-empty bounded ordered set.
The last element in S is z.
Element z is removed from S.
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Definition - S is a bounded ordered set of whole numbers from 0 to q in the set of whole numbers, W, if and only if S is identical to T once the following procedure is done:
Procedure
T is an empty set.
W is the set of whole numbers.
Element q is a whole number in W.
AA. Element k, in W, is 0.
BB. Append k into T.
CC. If k is q, this procedure is done, and T is the ordered set of non-positive numbers in W from 0 to q.
DD. The non-positive number next in order from k, in W, is the element of W that k becomes now.
EE. Append k into T.
FF. If k is not q, goto CC; otherwise this procedure is done, and T is the ordered set of non-positive numbers in W from 0 to q.
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Definition - S is a bounded ordered set whole of numbers from p to q in the set of whole numbers, W, if and only if each following statement is true:
W is the set of whole numbers.
Elements p and q are in W.
Bounded ordered set K is the set of of whole numbers from 0 to p, in W.
Bounded ordered set E is the set resulting when p is removed from K.
Bounded ordered set J is the set of whole numbers from 0 to q, in W.
S is the set of elements of set J that are not in set E.
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Definition - Item j is a positive number if and only if j is a whole number and j is not zero.
Definition - P is the set of positive numbers if and only if each element in P is a whole number and each element in P is not zero.
Definition - Whole number B is greater than whole number A if and only if the set of whole numbers from 0 to B, excluding B, is a set that is not empty and it contains A as an element.
Definition - Whole number A is less than whole number B if and only if whole number B is greater than whole number A.
Notation - A is greater than B may be written A > B. A is less than B may be written A < B.
Definition - Whole number B is greater than whole number A by positive number T if and only if the bounded ordered set of whole numbers, in W, from A to B has a one-to-one correspondence with the bounded ordered set of whole numbers, in W, from 0 to T.
Definition - Whole number B is greater than or equal to whole number A by whole number T if and only if the bounded ordered set of whole numbers, in W, from A to B has a one-to-one correspondence with the bounded ordered set of whole numbers, in W, from 0 to T.
The conception of negative numbers arose from necessities in accounting and physics. Many other uses for negative numbers then became apparent.
In accounting, debts and deficits need to be quantified. In physics, the concept of a dimension requires requires a pair of directions, one quantified by positive numbers and the opposite direction quantified by other numbers, now called negative numbers.
In the development of the mathematical foundations for such numbers there is an important constraint. It is necessary to define a set of negative numbers as an succession of individually separate, distinct and discrete numbers beginning at zero. This set cannot be defined as a succession of numbers where the succession begins with some other number, and ends at zero.
This means that a positive number which is next in order from another positive number, is greater than that number, while negative numbers must be defined such that a negative number which is next in order from another negative number, is more negative than that number, and therefore less than that negative number. While being a greater debt, it is a lesser amount in your bank balance.
Now, this casual description of goals needs to be followed by a logical and rigorous development of negative numbers, as though none of the above had been said.
Definition - M is the set of non-positive numbers if and only if each following statement is true for any positive number H, in W:
W is the unbounded ordered set of whole numbers.
M is an ordered set.
Each element in M is represented in writing by a text element that is unique among the text items representing elements in M.
The first element of M is zero, and is represented by the text item 0.
Each element, in M, other than zero, is represented by a text item of the form Cx, where C is a hyphen, and x is a text item representing a positive whole number in W.
The order of elements in the set M is the order of the text items which represent them, as specified in the following procedure. The procedure ends when the name of the amount of elements whose representations have been specified, is the positive number H.
Procedure
L0. W is the unbounded ordered set of whole numbers.
L1. C is a hyphen.
L2. H is a positive number in W.
L3. Element x is the first element of W, which is zero, represented in writing by the text item 0.
L4. Element y is the first element of M, which is zero, represented in writing by the text item 0.
L5. If H is next in order from x, in W, M is the set of non-positive numbers, and this procedure is done.
L6. The whole number next in order from x, in W, is the element in W that x now becomes.
L7. The element of M that is next in order from y, is represented by Cx, where x is the text item representing the positive number x in W.
L8. Go to L5.
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Definition - S is a bounded ordered set of non-positive numbers in M from 0 to q if and only if S is identical to T once the following procedure is done:
Procedure
T is an empty set.
M is the set of non-positive numbers.
Element q is a non-positive number in M.
AA. Element k, in M, is 0.
BB. Append k into T.
CC. If k is q, this procedure is done, and T is the ordered set of non-positive numbers in M from 0 to q.
DD. The non-positive number next in order from k, in M, is the element of M that k becomes now.
EE. Append k into T.
FF. If k is not q, goto CC; otherwise this procedure is done, and T is the ordered set of non-positive numbers in M from 0 to q.
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Definition - S is a bounded ordered set non-positive numbers from p to q, if and only if each following statement is true:
M is the set of non-positive numbers.
Elements p and q are in M.
Bounded ordered set K is the set of of non-positive numbers from 0 to p, in M.
Bounded ordered set E is the set resulting when p is removed from K.
Bounded ordered J is the set of non-positive numbers from 0 to q, in M.
S is the set of elements of set J that are not in set E.
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Definition - Z is the set of integers if and only if each following statement is true:
W is the set of whole numbers.
M is the set of non-positive numbers.
Z is the union of W and M.
The intersection of W and M is a set containing only zero.
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Definition - Element n is a negative integer if and only if n is a non-positive number and n is not zero.
Definition - U is the set of negative integers if and only if every non-positive number in the set of integers is in U except zero, and U contains no other elements.
Definition - The absolute value of any integer, p, is Q if and only if Q is the name of the amount of elements in the set of integers between zero and p, excluding zero.
Notation - The absolute value, whether p is positive, negative, or zero, is written either as abs(p) or |p|.
Definition - S is the set of integers between 0 and p if and only if the statements in one of the following cases are true:
Case I.
Item p is a non-positive number.
S contains exactly those elements, in M, that are in the set of non-positive numbers from 0 to p, in M.
Case II.
Item p is a whole number.
S contains exactly those elements, in W, that are in the set of whole numbers from 0 to p, in W.
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Definition - Integer B is greater than integer A if and only if M is the ordered set of non-positive numbers, and the statements in any of the following cases are true:
Case A
A and B are both negative integers, and B is in the bounded ordered set of non-positive numbers, in M, from -1 to A, excluding A.
Case B
A and B are both positive integers, and A is in the bounded ordered set of whole numbers, in W, from 1 to B, excluding B.
Case C
A is a non-positive number, and B is a positive integer, or A is a negative integer and B is a whole number.
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Definition - Integer A is less than integer B if and only if B is greater than A.
There is a reason, or perhaps an excuse, for calling the elements of M, other than zero, negative numbers. They each negate some whole number, other than zero, in sums including both.
Definition - The sum of any pair of integers is the integer s if and only if W is the set of whole numbers, and M is the set of non-positive numbers, and the statements in any of the following cases are true:
Case A
The pair of integers is comprised of positive integers p and q; and the set in W from 0 to p has a one-to-one correspondence with the set of whole numbers from q to r, for some positive integer r, and s is r.
Case B
The pair of integers is comprised of negative integers p and q; and the set, in M, from 0 to p has a one-to-one correspondence with the set, in M, of integers from q to r, for some negative integer r, and s is r.
Case C
The pair of integers is comprised of p in W, and q in M; and the statements in either of the following subcases are true:
Subcase C.I
Whole number p is greater than or equal to |q| by whole number t, and s is t.
Subcase C.II
Whole number |q| is greater than p by positive number t, and s is the negative integer such that |s| = t.
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If we are to have a directional term that can be applied to sets of integers consistently, regardless of whether a set of numbers is from +5 to +20, or from -40 to -3, or from -12 to +8, then inconsistency and possible confusion is created if our only directional term is "next in order."
Traditionally, when integers are written on a number line, the order, left to right, shows the lesser quantities on the left and the greater quantities the right, whether the elements next in order in the component sets in the integers have "next in order" elements on the left or the right.
Therefore, the integers must be referred to with terms that are in addition to those that apply to its subsets. Their order can be, and should be, referred to in order of quantity or amount.
Here is an integer number line:
. . . -46, -45, -44, . . -4, -3, -2, -1, 0, 1, 2, 3, 4, . . . 44, 45, 46, . . .
Number -46 is next in order from -45 in M (the set of non-positive numbers) and less than -45, while +46 is next in order from +45 in W (the set of whole numbers) and greater than +45. These subsets of the integers are constructed as unending successions beginning at zero, and grow in opposite directions; but the amounts named by the numbers get less as one proceeds to the left, and greater as one proceeds to the right on an integer number line.
Definition - S is a bounded ordered set of integers from p to q if and only if the statements in one of the following three cases are true:
Case I.
Integer p is greater than integer q and the bounded ordered set S is empty.
Case II.
Integer p equals integer q and the bounded ordered set S contains only p = q.
Case III.
Integer q is greater than integer p and S is identical to R once the following procedure is done:
Procedure
R is an empty set.
M is the set of non-positive numbers.
W is the set of whole numbers.
V is a set that is sometimes M, and at all other times W.
AA. If p is a negative number, then M is the set that set V becomes now.
BB. If p is a whole number, then W is the set that V becomes now.
CC. Element t, in V, is p.
DD. Append t into set R.
EE. If V is M, then the element in V such that t is next in order from it, in V, is the element in V that u becomes now.
FF. If V is W, then the element in V that is next in order from t is the element in V that u becomes now.
GG. Append u into set R.
HH. Element u in R is next in order from element t in R.
JJ. If u is 0, then W is the set that V becomes now.
KK. If u is q, this procedure is done, and R is the bounded ordered set of integers from p to q.
LL. Item u is the element in V that t becomes now.
MM. Go to EE.
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Definition - Integer B subtracted from integer C is A if and only if the sum of A and B is C.
Notation C - B = A
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