Fraction: Difference between revisions

Latest revision as of 21:54, 15 August 2025

A fraction is a pair $p/q$ of integers $p,q:\mathbb {Z}$ , $q\neq 0$ , regarded from the perspective that $p/q=_{\mathbb {Q} }r/s$ if $ps=_{\mathbb {Z} }qr$ .

Ratio

I want to say something like: A ratio is an actual relationship between quantities.

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

specifically, there's something about the symbols that is the same as $Q$ .

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

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

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

Now, what about negative fractions?

[OLD stuff below as of 08/15/25]

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, which we denote $\mathbb {Q}$ , is concept of signed differences between ratios.

n-ratios, projective space

[TODO]

@@ Line 1: / Line 1: @@
-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.

Fraction: Difference between revisions

Latest revision as of 21:54, 15 August 2025

Ratio

Rational numbers

n-ratios, projective space

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