Integer - Revision history

Lfox at 21:15, 15 August 2025

2025-08-15T21:15:05Z

← Older revision		Revision as of 21:15, 15 August 2025
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	An '''integer''' is a difference between two multitudes, considered together as an [[Order\|ordered]] [[Set\|pair]]. The concept of integer is sometimes denoted by a symbol, <math>\mathbb{Z}</math>.		An '''integer''' is a difference between two multitudes, considered together as an [[Order\|ordered]] [[Set\|pair]]. The concept of integer is sometimes denoted by a symbol, <math>\mathbb{Z}</math>. [TODO old as of 08/15/25]

			As a ''noun'', an integer is a symbol like -2, -1, 0, 1, etc.

			As an ''adjective'', integers can be used to describe different things. For example, a signed difference -- see the rest of this article. Or, for example, a location on an oriented pointed line. Or, for example, a winding number.

	== Examples of integers ==		== Examples of integers ==

Lfox: /* Examples of integers */

2024-07-03T16:59:30Z

Examples of integers

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	* If B is the origin of the ray, and <math>L_A</math> is found to be 10 inches, then together they are a unit of the concept "-10." One may say that the signed distance between A and B is -10 inches.		* If B is the origin of the ray, and <math>L_A</math> is found to be 10 inches, then together they are a unit of the concept "-10." One may say that the signed distance between A and B is -10 inches.
	* If the distance between the origin and either point is not a whole number of inches, then the signed distance between them might not be an integer at all; it might be a [[rational number]].		* If the distance between the origin and either point is not a whole number of inches, then the signed distance between them might not be an integer at all; it might be a [[rational number]].
			Consider a point P on a line with an origin. Let <math>L</math> denote the number of centimeters between P and the origin which are to the ''left'' of the origin, and let <math>R</math> denote the number of centimeters between P and the origin which are to the ''right'' of the origin. Necessarily, either <math>L = 0</math> or <math>R = 0</math>. The tuple <math>(R,L)</math> is an integer (if L and R are integral), and it is the linear coordinate of P.

	== Difference ==		== Difference ==

Lfox at 16:21, 2 July 2024

2024-07-02T16:21:18Z

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	An '''integer''' is a ~~concept measuring the~~ difference between two multitudes, considered as ~~the two members of~~ an [[Order\|ordered]] [[Set\|pair]]. ~~In Objective Mathematics, the~~ concept of integer is ~~often~~ denoted by a symbol, <math>\mathbb{Z}</math>.		An '''integer''' is a difference between two multitudes, considered together as an [[Order\|ordered]] [[Set\|pair]]. The concept of integer is sometimes denoted by a symbol, <math>\mathbb{Z}</math>.

	== Examples of integers ==		== Examples of integers ==
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	* If the distance between the origin and either point is not a whole number of inches, then the signed distance between them might not be an integer at all; it might be a [[rational number]].		* If the distance between the origin and either point is not a whole number of inches, then the signed distance between them might not be an integer at all; it might be a [[rational number]].

	== ~~Differences~~ ==		== Difference ==
	The difference between two quantities A and B, is the quantity which would have to be added to the lesser quantity in order to make it equal to the greater quantity.		The '''difference''' between two quantities A and B, is the quantity which would have to be added to the lesser quantity in order to make it equal to the greater quantity. In the case where A and B are the same quantity, we say that the difference between them is zero, or 0.

	In the case where ~~there is no difference between~~ A and B, we say that the difference between them is zero, or 0.

	=== Examples ===		=== Examples ===

Lfox at 16:31, 18 April 2024

2024-04-18T16:31:12Z

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	== Negation of integers ==		== Negation of integers ==
	'''To negate''' an integer <math>n</math> is to reverse the ordering of the pair to which it refers. The negation of <math>n</math> is denoted by <math>-n</math>.		'''To negate''' an integer <math>n</math> is to reverse the ordering of the pair to which it refers. The negation of <math>n</math> is denoted by <math>-n</math>.

			The ordering is something that your consciousness added to the pair. So in some sense every integer (meaning every object referred to by an integer) carries a <math>\mathbb{Z}_2</math> [[symmetry]]. It's like a gauge symmetry (change in description) rather than a true symmetry.

	=== Examples ===		=== Examples ===

Lfox: /* Multiplication of integers */

2024-02-15T03:05:13Z

Multiplication of integers

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	In symbolic notation, <math>(n,m) + (a,b) := (n + a, m + b)</math>.		In symbolic notation, <math>(n,m) + (a,b) := (n + a, m + b)</math>.

			=== Examples ===
			[TODO put my drawing]

	== Negation of integers ==		== Negation of integers ==
	'''To negate''' an integer <math>n</math> is to reverse the ordering of the pair to which it refers. The negation of <math>n</math> is denoted by <math>-n</math>.		'''To negate''' an integer <math>n</math> is to reverse the ordering of the pair to which it refers. The negation of <math>n</math> is denoted by <math>-n</math>.

			=== Examples ===
			[TODO put my drawing]

	== Multiplication of integers ==		== Multiplication of integers ==
	~~The idea~~ is ~~that we will define~~<math>(n~~,m)~~\times ~~(p,q) := (np + mq, nq + mp)~~ </math>~~, but it is difficult to justify~~.		'''To multiply''' an integer <math>n</math> by a natural number <math>k</math>, is to add <math>n</math> to itself <math>k</math> times. The number measuring the resulting difference may be denoted as <math>kn</math> or <math>k \times n</math>.

	[TODO]		We can extend this to multiplication of integers by integers, as follows. To multiply <math>n</math> by 0 is 0. To multiply <math>n</math> by a negative integer, <math>-k</math>, is to add the negation of <math>n</math> to itself <math>k</math> times.

			In symbolic notation, we have described an operation<math>(n,m)\times (p,q) := (np + mq, nq + mp) </math>.

			=== Examples ===
			[TODO put tons and tons of examples here]

	== The traditional concept ==		== The traditional concept ==
	[TODO needs a lot of work]		[TODO needs a lot of work]

	The integers are sometimes taken as an irreducible primary in mathematics. Leopold Kronecker said~~[TODO citation needed]~~<blockquote>God made the integers; all ~~else~~ is the work of man. </blockquote>and I think this quote is famous because some mathematicians sympathize with it.		The integers are sometimes taken as an irreducible primary in mathematics. The mathematician Leopold Kronecker said<ref>Bell, E. T. ''Men of Mathematics''. Seventh printing, Simon & Schuster, 1937, p. 477.</ref><blockquote>God made the integers, all the rest is the work of man. </blockquote>and I think this quote is famous because some mathematicians sympathize with it.

	Integers are literally taken as an irreducible primary by ZFC set theory, which has the axiom of infinity (TODO write something else. Calling this the integers, and not---say, the natural numbers---is an oversimplification.)		Integers are literally taken as an irreducible primary by ZFC set theory, which has the axiom of infinity (TODO write something else. Calling this the integers, and not---say, the natural numbers---is an oversimplification.)
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	There's a construction I have seen called the Grothendieck construction, which is in some ways similar to Objective Mathematics' way of viewing the integers. The idea is to view the integers as the set <math>\mathbb{N} \times \mathbb{N} / \sim</math>, where <math>(n,m) \sim (p,q) </math> if there exists some <math>k \in \mathbb{N}</math> such that <math>(p,q) = (n + k, m + k)</math>.		There's a construction I have seen called the Grothendieck construction, which is in some ways similar to Objective Mathematics' way of viewing the integers. The idea is to view the integers as the set <math>\mathbb{N} \times \mathbb{N} / \sim</math>, where <math>(n,m) \sim (p,q) </math> if there exists some <math>k \in \mathbb{N}</math> such that <math>(p,q) = (n + k, m + k)</math>.

	I am sure that versions of it long ~~predates~~ Grothendieck, but it was made famous by Grothendieck because it's the starting point of K theory.		I am sure that versions of this construction long predate Grothendieck, but it was made famous by Grothendieck because it's the starting point of K theory.

			== References ==

Lfox: /* Examples of integers */

2024-02-03T22:56:25Z

Examples of integers

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	Consider two piles of apples. Suppose that the two piles are ordered, so that we may refer to one as "pile #1" and the other as "pile #2."		Consider two piles of apples. Suppose that the two piles are ordered, so that we may refer to one as "pile #1" and the other as "pile #2."

	* If pile #1 contains 3 apples, and pile #2 contains 5 apples, then together they are a unit of the concept "-2."		* If pile #1 contains 3 apples, and pile #2 contains 5 apples, then together they are a unit of the concept "-2." One may say that the signed difference between them is -2 apples.
	* If pile #1 contains 71 apples, and pile #2 contains 71 apples, then together they are a unit of the concept "0."		* If pile #1 contains 71 apples, and pile #2 contains 71 apples, then together they are a unit of the concept "0." One may say that the signed difference between them is 0 apples.
	* If pile #1 contains 3 apples, and pile #2 is empty (contains no apples), then together they are a unit of the concept "3."		* If pile #1 contains 3 apples, and pile #2 is empty (contains no apples), then together they are a unit of the concept "3." One may say that the signed difference between them is 3 apples.
	Consider two sets of gold coins. "Set #2" is the set of gold coins which Bob has withdrawn from his bank account. "Set #1" is the set of gold coins which Bob has deposited in his bank account.		Consider two sets of gold coins. "Set #2" is the set of gold coins which Bob has withdrawn from his bank account. "Set #1" is the set of gold coins which Bob has deposited in his bank account.

	* If set #1 contains 6 gold coins, and set #2 contains 5 gold coins, then together they are a unit of the concept "1." His balance is said to be 1 gold coin.		* If set #1 contains 6 gold coins, and set #2 contains 5 gold coins, then together they are a unit of the concept "1." His balance is said to be 1 gold coin.
	* If set #1 contains 100,000 gold coins, and set #2 contains 200,000 gold coins, then together they are a unit of the concept "-100,000." His balance is said to be -100,000 gold coins.		* If set #1 contains 100,000 gold coins, and set #2 contains 200,000 gold coins, then together they are a unit of the concept "-100,000." His balance is said to be -100,000 gold coins.
			Consider two [[Point\|points]] A and B on a [[Line\|ray]]. Let <math>L_A</math> denote the distance from A to the origin, and let <math>L_B</math> denote the distance from B to the origin. Choose the ordering of the set <math>\{ L_A, L_B \}</math> according to which <math>L_A</math> is first, and <math>L_B</math> is second. Then

			* If <math>L_A</math> is found to be 17 inches, and <math>L_B</math> is found to be 31 inches, then together they are a unit of the concept "14." One may say that the signed distance between A and B is 14 inches.
			* If B is the origin of the ray, and <math>L_A</math> is found to be 10 inches, then together they are a unit of the concept "-10." One may say that the signed distance between A and B is -10 inches.
			* If the distance between the origin and either point is not a whole number of inches, then the signed distance between them might not be an integer at all; it might be a [[rational number]].

	== Differences ==		== Differences ==

Lfox: /* Examples of integers */

2024-02-03T00:05:41Z

Examples of integers

← Older revision		Revision as of 00:05, 3 February 2024
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	== Examples of integers ==		== Examples of integers ==
	In this section, I will give examples of concretes subsumed under specific integers, like -2, 0, ~~and~~ 3.		In this section, I will give examples of concretes subsumed under specific integers, like -2, 0, 3, etc.

	Consider two piles of apples. Suppose that the two piles are ordered, so that we may refer to one as "pile #1" and the other as "pile #2."		Consider two piles of apples. Suppose that the two piles are ordered, so that we may refer to one as "pile #1" and the other as "pile #2."
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	* If pile #1 contains 71 apples, and pile #2 contains 71 apples, then together they are a unit of the concept "0."		* If pile #1 contains 71 apples, and pile #2 contains 71 apples, then together they are a unit of the concept "0."
	* If pile #1 contains 3 apples, and pile #2 is empty (contains no apples), then together they are a unit of the concept "3."		* If pile #1 contains 3 apples, and pile #2 is empty (contains no apples), then together they are a unit of the concept "3."
			Consider two sets of gold coins. "Set #2" is the set of gold coins which Bob has withdrawn from his bank account. "Set #1" is the set of gold coins which Bob has deposited in his bank account.

			* If set #1 contains 6 gold coins, and set #2 contains 5 gold coins, then together they are a unit of the concept "1." His balance is said to be 1 gold coin.
			* If set #1 contains 100,000 gold coins, and set #2 contains 200,000 gold coins, then together they are a unit of the concept "-100,000." His balance is said to be -100,000 gold coins.

	== Differences ==		== Differences ==

Lfox at 00:00, 3 February 2024

2024-02-03T00:00:36Z

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-An '''integer''' is a concept measuring the difference between two multitudes, considered as two members of an ordered set. In Objective Mathematics, the concept of integer is often denoted by a symbol, <math>\mathbb{Z}</math>.
+An '''integer''' is a concept measuring the difference between two multitudes, considered as the two members of an [[Order|ordered]] [[Set|pair]]. In Objective Mathematics, the concept of integer is often denoted by a symbol, <math>\mathbb{Z}</math>.
 == Differences ==
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 == Signed differences ==
 A difference is something that is identified with respect to two quantities. For a difference, the ''order'' of the two quantities does not matter; the difference between the smaller and the larger is the same as the difference between the larger and the smaller.
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 in fact there is a [[symmetry]] between them.

Lfox at 21:58, 2 February 2024

2024-02-02T21:58:05Z

← Older revision		Revision as of 21:58, 2 February 2024
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	An '''integer''' is a ~~signed~~ difference between ~~[[Natural numbers\|whole quantities]]~~. In Objective Mathematics, the concept of integer is often denoted by a symbol, <math>\mathbb{Z}</math>.		An '''integer''' is a concept measuring the difference between two multitudes, considered as two members of an ordered set. In Objective Mathematics, the concept of integer is often denoted by a symbol, <math>\mathbb{Z}</math>.

	== Differences ==		== Differences ==
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	== Signed differences ==		== Signed differences ==
	~~[TODO I'm probably going to rewrite this entire section]~~

	A difference is something that is identified with respect to two quantities. For a difference, the ''order'' of the two quantities does not matter; the difference between the smaller and the larger is the same as the difference between the larger and the smaller.		A difference is something that is identified with respect to two quantities. For a difference, the ''order'' of the two quantities does not matter; the difference between the smaller and the larger is the same as the difference between the larger and the smaller.

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	=== Examples ===		=== Examples ===
			The difference between a pile consisting of 3 apples, and a pile consisting of 5 apples, is 2 apples.

			in fact there is a [[symmetry]] between them.

Lfox: /* Non-examples */

2024-01-22T00:59:34Z

Non-examples

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	The difference between a pile consisting of 3 apples, and a pile consisting of 5 apples, is 2 apples.		The difference between a pile consisting of 3 apples, and a pile consisting of 5 apples, is 2 apples.

	The difference between 3 and 5 is 2.		The difference between a pile consisting of 5 apples, and a pile consisting of 3 apples, is 2 apples.

			The difference between 3 and 5 is 2.

	=== Non-examples ===		=== Non-examples ===
	"A difference between me and my friend is that I like chocolate ice cream, but he doesn't." This is a perfectly valid use of the concept of difference, but it's ~~not in a mathematical setting because it is~~ not a difference of quantities.		"A difference between me and my friend is that I like chocolate ice cream, but he doesn't." This is a perfectly valid use of the concept of difference, but it's not a difference of quantities.

	== Signed differences ==		== Signed differences ==
			[TODO I'm probably going to rewrite this entire section]

	A difference is something that is identified with respect to two quantities. For a difference, the ''order'' of the two quantities does not matter; the difference between the smaller and the larger is the same as the difference between the larger and the smaller.		A difference is something that is identified with respect to two quantities. For a difference, the ''order'' of the two quantities does not matter; the difference between the smaller and the larger is the same as the difference between the larger and the smaller.