Fraction: Difference between revisions
(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.") |
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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 == | |||
A ratio is [TODO] | |||
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 | Rational numbers, which we denote <math>\mathbb{Q}</math>, is concept of signed differences between ratios. | ||
== n-ratios, projective space == | |||
[TODO] | |||
Latest revision as of 03:54, 12 March 2025
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 or . As soon as you take a quotient, you "lose" that information.
Ratio
A ratio is [TODO]
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 , is concept of signed differences between ratios.
n-ratios, projective space
[TODO]