Real number: Difference between revisions
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I am not yet sure how to define real numbers. Many irrational numbers (e.g. <math>\sqrt{2}</math> and <math>\pi</math>) are in fact valid concepts, but the standard definition of the reals involves infinite nonsense. | I am not yet sure how to define real numbers. Many irrational numbers (e.g. <math>\sqrt{2}</math> and <math>\pi</math>) are in fact valid concepts, but the standard definition of the reals involves infinite nonsense. | ||
The concept of real numbers is a ''concept'': it identifies things out in reality. In particular, it does not need to be "constructed" via some set-theoretic method like Dedekind cuts, nor do such constructions even make sense. That any real number can be approximated by fractions is obvious: a fraction is the outcome of directly measuring any quantity with a standard ruler. | |||
A real number should really just be the concept of a ''number'', meant in the broadest sense. Strictly speaking, quaternions, complex numbers, elements of finite fields, etc. are not numbers. | |||
== Examples == | == Examples == | ||
Any [[fraction]]. | Any [[fraction]]. | ||
Some [[Radical|algebraic numbers]], like <math>\sqrt{2}</math>. | |||
[[Pi]] | [[Pi]] | ||
[[e]] | [[e]] | ||
The number <math>0.x_1 x_2 x_3 x_4 \cdots</math>, where <math>x_i : \{0,1\}</math> is its <math>i</math>th digit in base 2, and where <math>x_i = 1</math> if <math>i</math> is prime, and 0 otherwise. | The number <math>0.x_1 x_2 x_3 x_4 \cdots</math>, where <math>x_i : \{0,1\}</math> is its <math>i</math>th digit in base 2, and where <math>x_i = 1</math> if <math>i</math> is prime, and 0 otherwise. [TODO wtf does that mean] | ||
The Euler-Mascheroni constant | The Euler-Mascheroni constant (where we don't actually know for sure whether or not it is rational) [TODO I'm not sure if it really makes sense] | ||
Latest revision as of 18:52, 8 July 2024
I am not yet sure how to define real numbers. Many irrational numbers (e.g. and ) are in fact valid concepts, but the standard definition of the reals involves infinite nonsense.
The concept of real numbers is a concept: it identifies things out in reality. In particular, it does not need to be "constructed" via some set-theoretic method like Dedekind cuts, nor do such constructions even make sense. That any real number can be approximated by fractions is obvious: a fraction is the outcome of directly measuring any quantity with a standard ruler.
A real number should really just be the concept of a number, meant in the broadest sense. Strictly speaking, quaternions, complex numbers, elements of finite fields, etc. are not numbers.
Examples
Any fraction.
Some algebraic numbers, like .
The number , where is its th digit in base 2, and where if is prime, and 0 otherwise. [TODO wtf does that mean]
The Euler-Mascheroni constant (where we don't actually know for sure whether or not it is rational) [TODO I'm not sure if it really makes sense]