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. | ||
I don't ''need'' to give them a set-theoretic definition. Real numbers are basically just the concept of numbers. | |||
== Examples == | == Examples == | ||
Any [[fraction]]. | Any [[fraction]]. | ||
Any [[Radical|algebraic number]] like <math>\sqrt{2}</math>. | Any [[Radical|algebraic number]], like <math>\sqrt{2}</math>. | ||
[[Pi]] | [[Pi]] | ||
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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. | ||
The Euler-Mascheroni constant | The Euler-Mascheroni constant (where we don't actually know for sure whether or not it is rational) | ||
Revision as of 21:35, 18 April 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.
I don't need to give them a set-theoretic definition. Real numbers are basically just the concept of numbers.
Examples
Any fraction.
Any algebraic number, like .
The number , where is its th digit in base 2, and where if is prime, and 0 otherwise.
The Euler-Mascheroni constant (where we don't actually know for sure whether or not it is rational)