Fraction - Revision history

Lfox at 21:54, 15 August 2025

2025-08-15T21:54:23Z

← Older revision		Revision as of 21:54, 15 August 2025
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	A '''fraction''' is a ~~concept which measures magnitude~~.		A '''fraction''' is a pair <math>p/q</math> of integers <math>p,q : \mathbb{Z} </math>, <math>q \neq 0</math>, regarded from the perspective that <math>p / q =_{\mathbb{Q}} r / s </math> if <math>p s =_{\mathbb{Z}} q r</math>.

	Is 3/~~4 = 6~~/8? Well ~~yes~~, in the ~~obvious sense~~. ~~But no~~, in the ~~sense that 3~~/~~4 means that something was divided into four pieces~~ and we ~~are considering 3~~ of ~~them~~, ~~whereas 6/8ths means~~ that ~~something was divided into 8 pieces and we are considering 6~~ of ~~them~~. ~~Those~~ two ~~situations~~ are ~~not literally equivalent~~.		== Ratio ==
			I want to say something like: A ratio is an actual relationship between quantities.

			Suppose that <math>p/q > 0</math>, and that it is describing some quantity <math>Q</math>.

			specifically, there's something about the ''symbols'' that is the same as <math>Q</math>.

			is <math>Q</math> a quantity? or is it a relationship between quantities? Well it could be either: we could have 3/5ths of a pizza, OR: we could have the ratio of the circumference of a pizza to its diameter. The former has units, the latter doesn't. The former is a quantity, and as for the latter... I guess it is not a quantity? Yeah. It's a quantitative relationship, but not a quantity.

			Let's stick with the former case, where <math>Q</math> is an actual quantity. What then does it mean to identify <math>Q</math> with <math>p/q</math>? Well first of all, there has to be some other quantity <math>\overline{Q}</math> that is being used as a foil. And then the idea is that <math>q</math> <math>Q</math>s would be the same quantity as <math>p</math> <math>\overline{Q}</math>s.

			Now, for the latter case, suppose we have two quantities <math>Q</math> and <math>\overline{Q}</math>. To understand the relationship between these quantities, we understand it in terms of sameness. If <math>q</math> <math>Q</math>s is the same quantity as <math>p</math> <math>\overline{Q}</math>s, then we say that the ratio of <math>Q</math> to <math>\overline{Q}</math> is <math>p/q</math>.

			So really these two cases are the same. We can say either that some quantity is <math>p/q</math>ths of another quantity, or else we can say that the ratio of the first to the second is <math>p/q</math>.

	~~If you want to retain that information~~, ~~you should identify whatever you are seeing as a pair, like <math>(3,4)</math> or <math>(6,8)</math>. As soon as you take a quotient, you "lose" that information.~~		Now, what about negative fractions?

	~~== Ratio ==~~		[OLD stuff below as of 08/15/25]
	~~A ratio is~~ [~~TODO~~]

	Ratios are very fundamental. Any measurement at all involves ratios.		Ratios are very fundamental. Any measurement at all involves ratios.

Lfox: /* Ratio */

2025-03-12T03:54:11Z

Ratio

← Older revision		Revision as of 03:54, 12 March 2025
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	== Rational numbers ==		== Rational numbers ==
	Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between ratios.		Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between ratios.

			== n-ratios, projective space ==
			[TODO]

Lfox at 02:11, 21 April 2024

2024-04-21T02:11:21Z

← Older revision		Revision as of 02:11, 21 April 2024
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	A '''fraction''' is a concept which measures magnitude.		A '''fraction''' is a concept which measures magnitude.

			Is 3/4 = 6/8? Well yes, in the obvious sense. But no, in the sense that 3/4 means that something was divided into four pieces and we are considering 3 of them, whereas 6/8ths means that something was divided into 8 pieces and we are considering 6 of them. Those two situations are not literally equivalent.

			If you want to retain that information, you should identify whatever you are seeing as a pair, like <math>(3,4)</math> or <math>(6,8)</math>. As soon as you take a quotient, you "lose" that information.

	== Ratio ==		== Ratio ==

Lfox: /* Ratio */

2024-04-18T21:40:44Z

Ratio

← Older revision		Revision as of 21:40, 18 April 2024
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	Ratios are very fundamental. Any measurement at all involves ratios.		Ratios are very fundamental. Any measurement at all involves ratios.

			A natural number is a set of several things, thought of in relation to a single thing.

			A fraction is a set of one or more things, thought of in relation to a set of one or more things.

			Just like how I defined an integer as an ordered pair of natural numbers, I could define a fraction as an ordered pair of natural numbers. The difference is that these pairs have a different equivalence relation, and different addition / multiplication operations.

	== Rational numbers ==		== Rational numbers ==
	Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between ratios.		Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between ratios.

Lfox at 20:10, 17 April 2024

2024-04-17T20:10:34Z

← Older revision		Revision as of 20:10, 17 April 2024
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	A '''fraction''' is a concept which measures magnitude.		A '''fraction''' is a concept which measures magnitude.

			== Ratio ==
			A ratio is [TODO]

			Ratios are very fundamental. Any measurement at all involves ratios.

	== Rational numbers ==		== Rational numbers ==
	Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between ~~fractions~~.		Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between ratios.

Lfox: Created page with "A '''fraction''' is a concept which measures magnitude. == Rational numbers == Rational numbers, which we denote $\mathbb{Q}$ , is concept of signed differences between fractions."

2024-01-27T23:35:31Z

Created page with "A '''fraction''' is a concept which measures magnitude. == Rational numbers == Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between fractions."

New page

A '''fraction''' is a concept which measures magnitude.

== Rational numbers ==
Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between fractions.

Fraction - Revision history

Lfox at 21:54, 15 August 2025

Lfox: /* Ratio */

Lfox at 02:11, 21 April 2024

Lfox: /* Ratio */

Lfox at 20:10, 17 April 2024

Lfox: Created page with "A '''fraction''' is a concept which measures magnitude. == Rational numbers == Rational numbers, which we denote \mathbb{Q}, is concept of signed differences between fractions."

Lfox: Created page with "A '''fraction''' is a concept which measures magnitude. == Rational numbers == Rational numbers, which we denote $\mathbb{Q}$ , is concept of signed differences between fractions."